Search: a324847 -id:a324847
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A324850
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Numbers divisible by the product of their prime indices.
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+10
71
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1, 2, 4, 6, 8, 12, 16, 24, 28, 30, 32, 36, 48, 56, 60, 64, 72, 96, 112, 120, 128, 144, 152, 156, 168, 180, 192, 216, 224, 240, 256, 288, 304, 312, 330, 336, 360, 384, 432, 448, 476, 480, 512, 576, 608, 624, 660, 672, 720, 768, 784, 828, 840, 848, 864, 888, 896
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OFFSET
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1,2
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COMMENTS
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A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798, with product A003963(n). For example, the prime indices of 30 are {1,2,3}, with product 6, which divides 30, so 30 is in the sequence.
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LINKS
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FORMULA
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EXAMPLE
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The sequence of terms together with their prime indices begins:
1: {}
2: {1}
4: {1,1}
6: {1,2}
8: {1,1,1}
12: {1,1,2}
16: {1,1,1,1}
24: {1,1,1,2}
28: {1,1,4}
30: {1,2,3}
32: {1,1,1,1,1}
36: {1,1,2,2}
48: {1,1,1,1,2}
56: {1,1,1,4}
60: {1,1,2,3}
64: {1,1,1,1,1,1}
72: {1,1,1,2,2}
96: {1,1,1,1,1,2}
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MATHEMATICA
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Select[Range[100], Divisible[#, Times@@Cases[If[#==1, {}, FactorInteger[#]], {p_, k_}:>PrimePi[p]^k]]&]
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PROG
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(PARI) isok(n) = my(f=factor(n)); !(n % prod(k=1, #f~, primepi(f[k, 1])^f[k, 2])); \\ Michel Marcus, Mar 22 2019
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CROSSREFS
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Cf. A003963, A036844, A120383, A324847, A324756, A324758, A324847, A324848, A324849, A324852, A324853, A324856, A324923, A324924, A324925, A324931.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A047967
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Number of partitions of n with some part repeated.
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+10
61
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0, 0, 1, 1, 3, 4, 7, 10, 16, 22, 32, 44, 62, 83, 113, 149, 199, 259, 339, 436, 563, 716, 913, 1151, 1453, 1816, 2271, 2818, 3496, 4309, 5308, 6502, 7959, 9695, 11798, 14298, 17309, 20877, 25151, 30203, 36225, 43323, 51748, 61651, 73359, 87086, 103254, 122164
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OFFSET
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0,5
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COMMENTS
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Also number of partitions of n with at least one even part. - Vladeta Jovovic, Sep 10 2003. Example: a(5)=4 because we have [4,1], [3,2], [2,2,1] and [2,1,1,1] ([5], [3,1,1] and [1,1,1,1,1] do not qualify). - Emeric Deutsch, Mar 30 2006
Also number of partitions of n (where it is assumed that the least part is 0) such that at least one difference is at least two. Example: a(5)=4 because we have [5,0], [4,1,0], [3,2,0] and [3,1,1,0] ([2,2,1,0], [2,1,1,1,0] and [1,1,1,1,1,0] do not qualify). - Emeric Deutsch, Mar 30 2006
The Heinz numbers of these partitions (with some part repeated) are given by A013929. Equivalent to Vladeta Jovovic's comment, a(n) is also the number of integer partitions whose product of parts is even. The Heinz numbers of these latter partitions are given by A324929. - Gus Wiseman, Mar 23 2019
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LINKS
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FORMULA
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G.f.: Sum_{k>=1} x^(2*k)*(Product_{j>=k+1} (1+x^j)) / Product_{j=1..k} (1-x^j) = Sum_{k>=1} x^(2*k)/(Product_{j=1..2*k} (1-x^j)*Product_{j>=k} (1-x^(2*j+1))). - Emeric Deutsch, Mar 30 2006
G.f.: 1/P(x) - P(x^2)/P(x) where P(x) = Product_{k>=1} (1-x^k). - Joerg Arndt, Jun 21 2011
a(n) = p(n-2)+p(n-4)-p(n-10)-p(n-14)+...+(-1^(j-1))*p(n-j*(3*j-1)) + (-1^(j-1))*p(n-j*(3*j+1))+..., where p(n) = A000041(n). - Gregory L. Simay, Aug 28 2023
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EXAMPLE
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a(5) = 4 because we have [3,1,1], [2,2,1], [2,1,1,1] and [1,1,1,1,1] ([5], [4,1] and [3,2] do not qualify).
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MAPLE
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g:=sum(x^(2*k)*product(1+x^j, j=k+1..70)/product(1-x^j, j=1..k), k=1..40): gser:=series(g, x=0, 50): seq(coeff(gser, x, n), n=0..44); # Emeric Deutsch, Mar 30 2006
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MATHEMATICA
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Table[PartitionsP[n]-PartitionsQ[n], {n, 0, 50}] (* Harvey P. Dale, Jan 17 2019 *)
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PROG
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(PARI) x='x+O('x^66); concat([0, 0], Vec(1/eta(x)-eta(x^2)/eta(x))) \\ Joerg Arndt, Jun 21 2011
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A120383
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A number n is included if it satisfies: m divides n for all m's where the m-th prime divides n.
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+10
50
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1, 2, 4, 6, 8, 12, 16, 18, 24, 28, 30, 32, 36, 48, 54, 56, 60, 64, 72, 78, 84, 90, 96, 108, 112, 120, 128, 144, 150, 152, 156, 162, 168, 180, 192, 196, 216, 224, 234, 240, 252, 256, 270, 288, 300, 304, 312, 324, 330, 336, 360, 384, 390, 392, 414, 420, 432, 444, 448
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OFFSET
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1,2
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COMMENTS
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If n is in the sequence, then 2*n is also in the sequence.
a(2) = 2 is the only prime number in the sequence.
a(1) = 1 is the only odd number in the sequence.
(End)
Numbers divisible by all of their prime indices. A prime index of n is a number m such that prime(m) divides n. For example, the prime indices of 78 = prime(1) * prime(2) * prime(6) are {1,2,6}, all of which divide 78, so 78 is in the sequence. - Gus Wiseman, Mar 23 2019
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LINKS
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EXAMPLE
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28 = 2^2 * 7. 2 is the first prime, 7 is the 4th prime. Since 1 and 4 both divide 28, then 28 is included in the sequence.
78 = 2 * 3 * 13. 2 is the first prime, 3 is the 2nd prime and 13 is the 6th prime. Since 1 and 2 and 6 each divide 78, then 78 is in the sequence. (Note that 1 * 2 * 6 does not divide 78.)
The sequence of terms together with their prime indices begins:
1: {}
2: {1}
4: {1,1}
6: {1,2}
8: {1,1,1}
12: {1,1,2}
16: {1,1,1,1}
18: {1,2,2}
24: {1,1,1,2}
28: {1,1,4}
30: {1,2,3}
32: {1,1,1,1,1}
36: {1,1,2,2}
48: {1,1,1,1,2}
54: {1,2,2,2}
56: {1,1,1,4}
60: {1,1,2,3}
64: {1,1,1,1,1,1}
(End)
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MAPLE
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A000040inv := proc(n) local i; i:=1 ; while true do if ithprime(i) = n then RETURN(i) ; fi ; i := i+1 ; end ; end: isA120383 := proc(n) local pl, p, i, j ; pl := ifactors(n) ; pl := pl[2] ; for i from 1 to nops(pl) do p := pl[i] ; j := A000040inv(p[1]) ; if n mod j <> 0 then RETURN(false) ; fi ; od ; RETURN(true) ; end: for n from 2 to 800 do if isA120383(n) then printf("%d, ", n); fi ; od ; # R. J. Mathar, Sep 02 2006
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MATHEMATICA
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{1}~Join~Select[Range[2, 450], Function[n, AllTrue[PrimePi /@ FactorInteger[n][[All, 1]], Mod[n, #] == 0 &]]] (* Michael De Vlieger, Mar 24 2019 *)
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PROG
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(PARI) ok(n) = my (f=factor(n)); for (i=1, #f~, if (n % primepi(f[i, 1]), return (0))); return (1) \\ Rémy Sigrist, Apr 08 2017
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CROSSREFS
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Cf. A000720, A003963, A056239, A112798, A323440, A324846, A324847, A324848, A324850, A324852, A324856.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A324846
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Positive integers divisible by none of their prime indices.
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+10
25
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1, 3, 5, 7, 9, 11, 13, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 47, 49, 51, 53, 57, 59, 61, 63, 65, 67, 69, 71, 73, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 107, 109, 111, 113, 115, 117, 121, 123, 125, 127, 129, 131, 133, 137
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OFFSET
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1,2
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COMMENTS
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A prime index of n is a number m such that prime(m) divides n. For example, the prime indices of 5673 are {2,11,18}, none of which divides 5673, so 5673 belongs to the sequence.
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LINKS
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EXAMPLE
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The sequence of terms together with their prime indices begins:
1: {}
3: {2}
5: {3}
7: {4}
9: {2,2}
11: {5}
13: {6}
17: {7}
19: {8}
21: {2,4}
23: {9}
25: {3,3}
27: {2,2,2}
29: {10}
31: {11}
33: {2,5}
35: {3,4}
37: {12}
39: {2,6}
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MAPLE
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q:= n-> ormap(i-> irem(n, numtheory[pi](i[1]))=0, ifactors(n)[2]):
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MATHEMATICA
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Select[Range[100], !Or@@Cases[If[#==1, {}, FactorInteger[#]], {p_, _}:>Divisible[#, PrimePi[p]]]&]
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PROG
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(PARI) isok(n) = {my(f = factor(n)[, 1]); for (k=1, #f, if (!(n % primepi(f[k])), return (0)); ); return (1); } \\ Michel Marcus, Mar 19 2019
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CROSSREFS
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Cf. A324695, A324741, A324743, A324756, A324758, A324765, A324848, A324849, A324850, A324852, A324853.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A324849
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Positive integers divisible by none of their prime indices > 1.
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+10
20
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1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 16, 17, 19, 20, 21, 22, 23, 25, 26, 27, 29, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 43, 44, 46, 47, 49, 50, 51, 52, 53, 57, 58, 59, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 73, 74, 76, 77, 79, 80, 81, 82, 83, 85, 86, 87
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OFFSET
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1,2
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COMMENTS
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A prime index of n is a number m such that prime(m) divides n.
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LINKS
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EXAMPLE
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The sequence of terms together with their prime indices begins:
1: {}
2: {1}
3: {2}
4: {1,1}
5: {3}
7: {4}
8: {1,1,1}
9: {2,2}
10: {1,3}
11: {5}
13: {6}
14: {1,4}
16: {1,1,1,1}
17: {7}
19: {8}
20: {1,1,3}
21: {2,4}
22: {1,5}
23: {9}
25: {3,3}
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MAPLE
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filter:= proc(n) andmap(t -> not ((n/numtheory:-pi(t))::integer), numtheory:-factorset(n) minus {2}) end proc:
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MATHEMATICA
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Select[Range[100], !Or@@Cases[If[#==1, {}, FactorInteger[#]], {p_, _}:>If[p==2, False, Divisible[#, PrimePi[p]]]]&]
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PROG
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(PARI) is(n) = my(f=factor(n)[, 1]~, idc=[]); for(k=1, #f, idc=concat(idc, [primepi(f[k])])); for(t=1, #idc, if(idc[t]==1, next); if(n%idc[t]==0, return(0))); 1 \\ Felix Fröhlich, Mar 21 2019
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CROSSREFS
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Cf. A324695, A324741, A324743, A324756, A324758, A324765, A324846, A324847, A324848, A324850, A324852, A324853, A324856.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A324851
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Numbers > 1 divisible by the sum of their prime indices.
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+10
19
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2, 4, 6, 12, 15, 16, 20, 30, 35, 36, 42, 48, 56, 88, 99, 112, 120, 126, 130, 135, 143, 144, 160, 162, 180, 192, 210, 216, 220, 221, 228, 231, 242, 250, 256, 270, 275, 280, 288, 297, 300, 308, 322, 330, 338, 360, 396, 400, 408, 429, 435, 440, 455, 468, 480, 493
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OFFSET
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1,1
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COMMENTS
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A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The sum of prime indices of n is A056239(n). For example, the prime indices of 99 are {2,2,5}, with sum 9, a divisor of 99, so 99 is in the sequence.
For any k>=2, let d be a divisor of k such that d > A056239(k). Then 2^(d-A056239(k))*k is in the sequence. Similarly if k is in the sequence with d = A056239(k), then 2^d*k is in the sequence. - Robert Israel, Mar 19 2019
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LINKS
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EXAMPLE
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The sequence of terms together with their prime indices begins:
2: {1}
4: {1,1}
6: {1,2}
12: {1,1,2}
15: {2,3}
16: {1,1,1,1}
20: {1,1,3}
30: {1,2,3}
35: {3,4}
36: {1,1,2,2}
42: {1,2,4}
48: {1,1,1,1,2}
56: {1,1,1,4}
88: {1,1,1,5}
99: {2,2,5}
112: {1,1,1,1,4}
120: {1,1,1,2,3}
126: {1,2,2,4}
130: {1,3,6}
135: {2,2,2,3}
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MAPLE
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filter:= proc(n) local t; n mod add(numtheory:-pi(t[1])*t[2], t=ifactors(n)[2]) = 0 end proc:
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MATHEMATICA
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Select[Range[2, 100], Divisible[#, Plus@@Cases[If[#==1, {}, FactorInteger[#]], {p_, k_}:>PrimePi[p]*k]]&]
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PROG
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(PARI) isok(n) = {my(f = factor(n)); (n!=1) && !(n % sum(k=1, #f~, primepi(f[k, 1])*f[k, 2])); } \\ Michel Marcus, Mar 19 2019
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CROSSREFS
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Cf. A324695, A324741, A324743, A324847, A324756, A324758, A324765, A324847, A324848, A324849, A324850, A324852.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A324844
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Number of unlabeled rooted trees with n nodes where the branches of no non-leaf branch of any terminal subtree form a submultiset of the branches of the same subtree.
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+10
17
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1, 1, 2, 3, 7, 13, 32, 71, 170, 406, 1002, 2469, 6204, 15644, 39871, 102116, 263325, 682079, 1775600, 4640220
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OFFSET
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1,3
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LINKS
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EXAMPLE
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The a(1) = 1 through a(6) = 13 rooted trees:
o (o) (oo) (ooo) (oooo) (ooooo)
((o)) ((oo)) ((ooo)) ((oooo))
(((o))) (o(oo)) (o(ooo))
(((oo))) (((ooo)))
((o)(o)) ((o)(oo))
(o((o))) ((o(oo)))
((((o)))) (o((oo)))
(oo((o)))
((((oo))))
(((o)(o)))
((o((o))))
(o(((o))))
(((((o)))))
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MATHEMATICA
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submultQ[cap_, fat_]:=And@@Function[i, Count[fat, i]>=Count[cap, i]]/@Union[List@@cap];
rallt[n_]:=Select[Union[Sort/@Join@@(Tuples[rallt/@#]&/@IntegerPartitions[n-1])], And@@Table[!submultQ[b, #], {b, DeleteCases[#, {}]}]&];
Table[Length[rallt[n]], {n, 10}]
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CROSSREFS
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The Matula-Goebel numbers of these trees are given by A324845.
Cf. A324694, A324738, A324744, A324749, A324754, A324759, A324765, A324768, A324838, A324843, A324846, A324847, A324848, A324849.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A324848
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Number of prime indices of n (counted with multiplicity) that divide n.
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+10
17
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0, 1, 0, 2, 0, 2, 0, 3, 0, 1, 0, 3, 0, 1, 1, 4, 0, 3, 0, 2, 0, 1, 0, 4, 0, 1, 0, 3, 0, 3, 0, 5, 0, 1, 0, 4, 0, 1, 0, 3, 0, 2, 0, 2, 1, 1, 0, 5, 0, 1, 0, 2, 0, 4, 1, 4, 0, 1, 0, 4, 0, 1, 0, 6, 0, 2, 0, 2, 0, 1, 0, 5, 0, 1, 2, 2, 0, 3, 0, 4, 0, 1, 0, 4, 0, 1, 0
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OFFSET
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1,4
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COMMENTS
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A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
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LINKS
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EXAMPLE
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The prime indices of 6776 are {1,1,1,4,5,5}, four of which {1,1,1,4} divide 6776, so a(6776) = 4.
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MATHEMATICA
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Table[Total[Cases[If[n==1, {}, FactorInteger[n]], {p_, k_}:>k/; Divisible[n, PrimePi[p]]]], {n, 100}]
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CROSSREFS
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The version for distinct prime indices is A324852.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A324843
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Number of unlabeled rooted trees with n nodes where the branches of any branch of any terminal subtree form a submultiset of the branches of the same subtree.
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+10
12
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1, 1, 1, 2, 2, 4, 4, 8, 9, 15, 17, 31, 35, 57, 70, 111, 136, 213, 265, 405, 517, 763, 987, 1458, 1893, 2736, 3611, 5161, 6836, 9702
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OFFSET
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1,4
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COMMENTS
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A subset of totally transitive rooted trees (A318185).
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LINKS
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EXAMPLE
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The a(1) = 1 through a(8) = 8 rooted trees:
o (o) (oo) (ooo) (oooo) (ooooo) (oooooo) (ooooooo)
(o(o)) (oo(o)) (oo(oo)) (ooo(oo)) (ooo(ooo))
(ooo(o)) (oooo(o)) (oooo(oo))
(o(o)(o)) (oo(o)(o)) (ooooo(o))
(oo(o)(oo))
(ooo(o)(o))
(o(o)(o)(o))
(o(o)(o(o)))
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MATHEMATICA
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submultQ[cap_, fat_]:=And@@Function[i, Count[fat, i]>=Count[cap, i]]/@Union[List@@cap];
rallt[n_]:=Select[Union[Sort/@Join@@(Tuples[rallt/@#]&/@IntegerPartitions[n-1])], And@@Table[submultQ[b, #], {b, #}]&];
Table[Length[rallt[n]], {n, 10}]
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CROSSREFS
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The Matula-Goebel numbers of these trees are given by A324842.
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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A324852
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Number of distinct prime indices of n that divide n.
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+10
11
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0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 2, 0, 3, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 1, 0, 1, 0, 2, 1, 2, 0, 1, 0, 3, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 1, 1, 0, 3, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 1
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OFFSET
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1,6
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COMMENTS
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A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
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LINKS
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EXAMPLE
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60060 has 7 prime indices {1,1,2,3,4,5,6}, all of which divide 60060, and 6 of which are distinct, so a(60060) = 6.
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MAPLE
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a:= n-> add(`if`(irem(n, numtheory[pi](i[1]))=0, 1, 0), i=ifactors(n)[2]):
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MATHEMATICA
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Table[Count[If[n==1, {}, FactorInteger[n]], {p_, _}/; Divisible[n, PrimePi[p]]], {n, 100}]
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PROG
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(PARI) a(n) = {my(f = factor(n)[, 1]); sum(k=1, #f, !(n % primepi(f[k]))); } \\ Michel Marcus, Mar 19 2019
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CROSSREFS
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The version for all prime indices (counted with multiplicity) is A324848.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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