Displaying 1-10 of 42 results found.
One plus the number of steps applying A325351 (Heinz number of augmented differences of reversed prime indices) to reach a fixed point.
+20
7
1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 2, 1, 1, 4, 1, 1, 1, 2, 1, 2, 1, 1, 3, 1, 1, 5, 1, 4, 2, 1, 1, 6, 1, 1, 4, 1, 1, 3, 1, 1, 1, 2, 2, 7, 1, 1, 2, 3, 1, 8, 1, 1, 3, 1, 1, 4, 1, 5, 5, 1, 1, 9, 4, 1, 2, 1, 1, 3, 1, 5, 6, 1, 1, 2, 1, 1, 4, 4, 1, 10, 1, 1, 3, 5, 1, 11, 1, 6, 1, 1, 2, 5, 2, 1, 7, 1, 1, 3
COMMENTS
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
The augmented differences aug(y) of an integer partition y of length k are given by aug(y)_i = y_i - y_{i + 1} + 1 if i < k and aug(y)_k = y_k. For example, aug(6,5,5,3,3,3) = (2,1,3,1,1,3).
EXAMPLE
Repeatedly applying A325351 starting with 78 gives 78 -> 66 -> 42 -> 30 -> 18 -> 12, and 12 is a fixed point, so a(78) = 6.
MATHEMATICA
primeptn[n_]:=If[n==1, {}, Reverse[Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]];
aug[y_]:=Table[If[i<Length[y], y[[i]]-y[[i+1]]+1, y[[i]]], {i, Length[y]}];
Table[Length[FixedPointList[Times@@Prime/@aug[primeptn[#]]&, n]]-1, {n, 50}]
PROG
(PARI)
augdiffs(n) = { my(diffs=List([]), f=factor(n), prevpi, pi=0, i=#f~); while(i, prevpi=pi; pi = primepi(f[i, 1]); if(prevpi, listput(diffs, 1+(prevpi-pi))); if(f[i, 2]>1, f[i, 2]--, i--)); if(pi, listput(diffs, pi)); Vec(diffs); };
A325351(n) = factorback(apply(prime, augdiffs(n)));
CROSSREFS
Cf. A056239, A093641, A112798, A130091, A289509, A325351, A325352, A325366, A325389, A325394, A325395, A325396.
Least gap in the partition having Heinz number n; index of the least prime not dividing n.
+10
55
1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 3
COMMENTS
The "least gap" of a partition is the least positive integer that is not a part of the partition. For example, the least gap of the partition [7,4,2,2,1] is 3.
We define the Heinz number of a partition p = [p_1, p_2, ..., p_r] as Product(p_j-th prime, j=1...r) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the partition [1, 1, 2, 4, 10] we get 2*2*3*7*29 = 2436. In the Maple program the subprogram B yields the partition with Heinz number n.
Sum of least gaps of all partitions of m = A022567(m). Index of the least prime not dividing n. (After a formula given by Heinz.)
Least k such that A002110(k) does not divide n. One more than the number of trailing zeros in primorial base representation of n, A049345. (End)
The least gap is also called the mex (minimal excludant) of the partition. - Gus Wiseman, Apr 20 2021
REFERENCES
G. E. Andrews and K. Eriksson, Integer Partitions, Cambridge Univ. Press, 2004, Cambridge.
Miklós Bóna, A Walk Through Combinatorics, World Scientific Publishing Co., 2002.
FORMULA
(End)
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1 + Sum_{k>=1} 1/ A002110(k) = 1.705230... (1 + A064648). - Amiram Eldar, Jul 23 2022
EXAMPLE
a(18) = 3 because the partition having Heinz number 18 = 2*3*3 is [1,2,2], having least gap equal to 3.
MAPLE
with(numtheory): a := proc (n) local B, q: B := proc (n) local nn, j, m: nn := op(2, ifactors(n)): for j to nops(nn) do m[j] := op(j, nn) end do: [seq(seq(pi(op(1, m[i])), q = 1 .. op(2, m[i])), i = 1 .. nops(nn))] end proc: for q while member(q, B(n)) = true do end do: q end proc: seq(a(n), n = 1 .. 150);
# second Maple program:
a:= n-> `if`(n=1, 1, (s-> min({$1..(max(s)+1)} minus s))(
{map(x-> numtheory[pi](x[1]), ifactors(n)[2])[]})):
# faster:
A257993 := proc(n) local p, c; c := 1; p := 2;
while n mod p = 0 do p := nextprime(p); c := c + 1 od: c end:
MATHEMATICA
Table[k = 1; While[! CoprimeQ[Prime@ k, n], k++]; k, {n, 100}] (* Michael De Vlieger, Jun 22 2017 *)
PROG
(Scheme)
(define ( A257993 n) (let loop ((n n) (i 1)) (let* ((p ( A000040 i)) (d (modulo n p))) (if (not (zero? d)) i (loop (/ (- n d) p) (+ 1 i)))))) (Python)
from sympy import nextprime, primepi
def a053669(n):
p = 2
while True:
if n%p!=0: return p
else: p=nextprime(p)
(PARI) a(n) = forprime(p=2, , if (n % p, return(primepi(p)))); \\ Michel Marcus, Jun 22 2017
CROSSREFS
The triangle counting partitions by this statistic is A264401. A maximal instead of minimal version is A339662. Positions of even terms are A342050. Positions of odd terms are A342051. A000070 counts partitions with a selected part.
A006128 counts partitions with a selected position.
A073491 lists numbers with gap-free prime indices.
A238709 counts partitions by sum and least difference.
A333214 lists positions of adjacent unequal prime gaps.
A339737 counts partitions by sum and greatest gap.
Cf. A001223, A001511, A005117, A026794, A029707, A072233, A079068, A098743, A124010, A279945, A325351, A325352.
Heinz number of the differences plus one of the integer partition with Heinz number n.
+10
31
1, 1, 1, 2, 1, 3, 1, 4, 2, 5, 1, 6, 1, 7, 3, 8, 1, 6, 1, 10, 5, 11, 1, 12, 2, 13, 4, 14, 1, 9, 1, 16, 7, 17, 3, 12, 1, 19, 11, 20, 1, 15, 1, 22, 6, 23, 1, 24, 2, 10, 13, 26, 1, 12, 5, 28, 17, 29, 1, 18, 1, 31, 10, 32, 7, 21, 1, 34, 19, 15, 1, 24, 1, 37, 6, 38
COMMENTS
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
The only fixed point is 1 because otherwise the sequence decreases omega ( A001222) by one.
EXAMPLE
The partition (3,2,2,1) with Heinz number 90 has differences plus one (2,1,2) with Heinz number 18, so a(90) = 18.
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
db[n_]:=Times@@Prime/@(1+Differences[primeMS[n]]);
Table[db[n], {n, 100}]
CROSSREFS
Cf. A007294, A049988, A056239, A093641, A112798, A240026, A320466, A325328, A325351, A325360, A325361, A325368, A325405.
Heinz numbers of partitions of crank 0.
+10
20
6, 10, 14, 22, 26, 34, 38, 46, 58, 62, 74, 82, 86, 94, 100, 106, 118, 122, 134, 140, 142, 146, 158, 166, 178, 194, 196, 202, 206, 214, 218, 220, 226, 254, 260, 262, 274, 278, 298, 300, 302, 308, 314, 326, 334, 340, 346, 358, 362, 364, 380, 382, 386, 394, 398
COMMENTS
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
See A257989 or the program for a definition of crank of a partition.
EXAMPLE
The sequence of terms together with their prime indices begins:
6: {1,2} 106: {1,16} 218: {1,29}
10: {1,3} 118: {1,17} 220: {1,1,3,5}
14: {1,4} 122: {1,18} 226: {1,30}
22: {1,5} 134: {1,19} 254: {1,31}
26: {1,6} 140: {1,1,3,4} 260: {1,1,3,6}
34: {1,7} 142: {1,20} 262: {1,32}
38: {1,8} 146: {1,21} 274: {1,33}
46: {1,9} 158: {1,22} 278: {1,34}
58: {1,10} 166: {1,23} 298: {1,35}
62: {1,11} 178: {1,24} 300: {1,1,2,3,3}
74: {1,12} 194: {1,25} 302: {1,36}
82: {1,13} 196: {1,1,4,4} 308: {1,1,4,5}
86: {1,14} 202: {1,26} 314: {1,37}
94: {1,15} 206: {1,27} 326: {1,38}
100: {1,1,3,3} 214: {1,28} 334: {1,39}
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
ck[y_]:=With[{w=Count[y, 1]}, If[w==0, Max@@y, Count[y, _?(#>w&)]-w]];
Select[Range[100], ck[primeMS[#]]==0&]
CROSSREFS
A000005 counts constant partitions.
A001522 counts partitions of positive crank.
A003242 counts anti-run compositions.
A064391 counts partitions by crank.
A064428 counts partitions of nonnegative crank.
A224958 counts compositions with alternating parts unequal.
A257989 gives the crank of the partition with Heinz number n.
Heinz numbers of integer partitions whose augmented differences are distinct.
+10
19
1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 17, 19, 21, 22, 23, 25, 26, 29, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 49, 51, 53, 57, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 99, 101, 102, 103
COMMENTS
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
The augmented differences aug(y) of an integer partition y of length k are given by aug(y)_i = y_i - y_{i + 1} + 1 if i < k and aug(y)_k = y_k. For example, aug(6,5,5,3,3,3) = (2,1,3,1,1,3).
The enumeration of these partitions by sum is given by A325349.
EXAMPLE
The sequence of terms together with their prime indices begins:
1: {}
2: {1}
3: {2}
5: {3}
6: {1,2}
7: {4}
9: {2,2}
10: {1,3}
11: {5}
13: {6}
14: {1,4}
17: {7}
19: {8}
21: {2,4}
22: {1,5}
23: {9}
25: {3,3}
26: {1,6}
29: {10}
31: {11}
MATHEMATICA
primeptn[n_]:=If[n==1, {}, Reverse[Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]];
aug[y_]:=Table[If[i<Length[y], y[[i]]-y[[i+1]]+1, y[[i]]], {i, Length[y]}];
Select[Range[100], UnsameQ@@aug[primeptn[#]]&]
CROSSREFS
Positions of squarefree numbers in A325351. Cf. A056239, A093641, A112798, A130091, A325349, A325355, A325367, A325368, A325389, A325394, A325395, A325396, A325405.
Heinz numbers of integer partitions whose augmented differences are weakly increasing.
+10
19
1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 15, 16, 17, 19, 23, 25, 27, 29, 31, 32, 35, 37, 41, 43, 47, 49, 53, 55, 59, 61, 64, 67, 71, 73, 75, 77, 79, 81, 83, 89, 91, 97, 101, 103, 105, 107, 109, 113, 119, 121, 125, 127, 128, 131, 137, 139, 143, 149, 151, 157, 163, 167
COMMENTS
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
The augmented differences aug(y) of an integer partition y of length k are given by aug(y)_i = y_i - y_{i + 1} + 1 if i < k and aug(y)_k = y_k. For example, aug(6,5,5,3,3,3) = (2,1,3,1,1,3).
The enumeration of these partitions by sum is given by A325356.
EXAMPLE
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}
11: {5}
13: {6}
15: {2,3}
16: {1,1,1,1}
17: {7}
19: {8}
23: {9}
25: {3,3}
27: {2,2,2}
29: {10}
31: {11}
32: {1,1,1,1,1}
MATHEMATICA
primeptn[n_]:=If[n==1, {}, Reverse[Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]];
aug[y_]:=Table[If[i<Length[y], y[[i]]-y[[i+1]]+1, y[[i]]], {i, Length[y]}];
Select[Range[100], OrderedQ[aug[primeptn[#]]]&]
CROSSREFS
Cf. A056239, A093641, A112798, A240026, A325351, A325356, A325360, A325362, A325366, A325389, A325395, A325396, A325400.
Numbers k which have an odd number of trailing zeros in their primorial base representation A049345(k).
+10
18
2, 4, 8, 10, 14, 16, 20, 22, 26, 28, 30, 32, 34, 38, 40, 44, 46, 50, 52, 56, 58, 60, 62, 64, 68, 70, 74, 76, 80, 82, 86, 88, 90, 92, 94, 98, 100, 104, 106, 110, 112, 116, 118, 120, 122, 124, 128, 130, 134, 136, 140, 142, 146, 148, 150, 152, 154, 158, 160, 164, 166, 170, 172, 176, 178, 180, 182, 184, 188, 190, 194, 196, 200, 202, 206, 208, 212
COMMENTS
Numbers k such that A276084(k) is odd. All the terms are even since odd numbers have 0 trailing zeros, and 0 is not odd.
The number of terms not exceeding A002110(m) for m>=1 is A002110(m) * Sum_{k=1..m}(-1)^k/ A002110(k) = 1, 2, 11, 76, 837, 10880, 184961, ... The asymptotic density of this sequence is Sum_{k>=1} (-1)^(k+1)/ A002110(k) = 0.362306... ( A132120). Also Heinz numbers of partitions with even least gap. The least gap (mex or minimal excludant) of a partition is the least positive integer that is not a part. The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions. - Gus Wiseman, Apr 23 2021 Numbers k such that A000720( A053669(k)) is even. Differences from the related A353531 seem to be terms that are multiples of 210, but not all of them, for example primorial 30030 (= 143*210) is in neither sequence. Consider also A038698. - Antti Karttunen, Apr 25 2022
EXAMPLE
2 is a term since A049345(2) = 10 has 1 trailing zero. 4 is a term since A049345(2) = 20 has 1 trailing zero. 30 is a term since A049345(2) = 1000 has 3 trailing zeros. The sequence of terms together with their prime indices begins:
2: {1} 46: {1,9} 90: {1,2,2,3}
4: {1,1} 50: {1,3,3} 92: {1,1,9}
8: {1,1,1} 52: {1,1,6} 94: {1,15}
10: {1,3} 56: {1,1,1,4} 98: {1,4,4}
14: {1,4} 58: {1,10} 100: {1,1,3,3}
16: {1,1,1,1} 60: {1,1,2,3} 104: {1,1,1,6}
20: {1,1,3} 62: {1,11} 106: {1,16}
22: {1,5} 64: {1,1,1,1,1,1} 110: {1,3,5}
26: {1,6} 68: {1,1,7} 112: {1,1,1,1,4}
28: {1,1,4} 70: {1,3,4} 116: {1,1,10}
30: {1,2,3} 74: {1,12} 118: {1,17}
32: {1,1,1,1,1} 76: {1,1,8} 120: {1,1,1,2,3}
34: {1,7} 80: {1,1,1,1,3} 122: {1,18}
38: {1,8} 82: {1,13} 124: {1,1,11}
40: {1,1,1,3} 86: {1,14} 128: {1,1,1,1,1,1,1}
44: {1,1,5} 88: {1,1,1,5} 130: {1,3,6}
(End)
MATHEMATICA
seq[max_] := Module[{bases = Prime@Range[max, 1, -1], nmax}, nmax = Times @@ bases - 1; Select[Range[nmax], OddQ @ LengthWhile[Reverse @ IntegerDigits[#, MixedRadix[bases]], #1 == 0 &] &]]; seq[4]
Select[Range[100], EvenQ[Min@@Complement[Range[PrimeNu[#]+1], PrimePi/@First/@FactorInteger[#]]]&] (* Gus Wiseman, Apr 23 2021 *)
PROG
(PARI)
A353525(n) = { for(i=1, oo, if(n%prime(i), return((i+1)%2))); }
k=0; n=0; while(k<77, n++; if(isA342050(n), k++; print1(n, ", "))); \\ Antti Karttunen, Apr 25 2022
CROSSREFS
The version for reversed binary expansion is A079523. Positions of even terms in A257993. A000070 counts partitions with a selected part.
A073491 lists numbers with gap-free prime indices.
A079067 counts gaps in prime indices.
A238709 counts partitions by sum and least difference.
A333214 lists positions of adjacent unequal prime gaps.
A339662 gives greatest gap in prime indices.
Cf. A000720, A001223, A005408, A026794, A029707, A038698, A047235, A079068, A121539, A286469, A286470, A325351, A353525 (characteristic function). Differs from A353531 for the first time at n=77, where a(77) = 212, as this sequence misses A353531(77) = 210.
Irregular triangle read by rows where row n lists the augmented differences of the reversed prime indices of n.
+10
18
1, 2, 1, 1, 3, 2, 1, 4, 1, 1, 1, 1, 2, 3, 1, 5, 2, 1, 1, 6, 4, 1, 2, 2, 1, 1, 1, 1, 7, 1, 2, 1, 8, 3, 1, 1, 3, 2, 5, 1, 9, 2, 1, 1, 1, 1, 3, 6, 1, 1, 1, 2, 4, 1, 1, 10, 2, 2, 1, 11, 1, 1, 1, 1, 1, 4, 2, 7, 1, 2, 3, 1, 2, 1, 1, 12, 8, 1, 5, 2, 3, 1, 1, 1
COMMENTS
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 augmented differences aug(q) of a (usually weakly decreasing) sequence q of length k are given by aug(q)_i = q_i - q_{i+1} + 1 if i < k and aug(q)_k = q_k. For example, we have aug(6,5,5,3,3,3) = (2,1,3,1,1,3).
One could argue that row n = 1 is empty, but adding it changes only the offset, not the data.
EXAMPLE
Triangle begins:
2: 1
3: 2
4: 1 1
5: 3
6: 2 1
7: 4
8: 1 1 1
9: 1 2
10: 3 1
11: 5
12: 2 1 1
13: 6
14: 4 1
15: 2 2
16: 1 1 1 1
For example, the reversed prime indices of 825 are (5,3,3,2), which have augmented differences (3,1,2,2).
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
aug[y_]:=Table[If[i<Length[y], y[[i]]-y[[i+1]]+1, y[[i]]], {i, Length[y]}];
Table[aug[Reverse[primeMS[n]]], {n, 30}]
CROSSREFS
Crossrefs found in the link are not repeated here.
Constant rows have indices A307824. The Heinz numbers of the rows are A325351.
0, 1, 2, 3, 4, 6, 8, 7, 5, 12, 16, 14, 32, 24, 10, 15, 64, 13, 128, 28, 20, 48, 256, 30, 9, 96, 11, 56, 512, 26, 1024, 31, 40, 192, 18, 29, 2048, 384, 80, 60, 4096, 52, 8192, 112, 22, 768, 16384, 62, 17, 25, 160, 224, 32768, 27, 36, 120, 320, 1536, 65536, 58, 131072, 3072, 44, 63, 72, 104, 262144, 448, 640, 50, 524288, 61, 1048576, 6144, 21
COMMENTS
Note the indexing: domain starts from one, while the range includes also zero. See also comments in A253564. The a(n)-th composition in standard order (graded reverse-lexicographic, A066099) is one plus the first differences of the weakly increasing sequence of prime indices of n with 1 prepended. See formula for a simplification. The triangular form is A358169. The inverse is A253565. Not prepending 1 gives A358171. For Heinz numbers instead of standard compositions we have A325351 (without prepending A325352). - Gus Wiseman, Dec 23 2022
FORMULA
As a composition of other permutations:
If 2n = Product_{i=1..k} prime(x_i) then a(n) = Sum_{i=1..k-1} 2^(x_k-x_{k-i}+i-1). - Gus Wiseman, Dec 23 2022
EXAMPLE
This represents the following bijection between partitions and compositions. The reversed prime indices of n together with the a(n)-th composition in standard order are:
1: () -> ()
2: (1) -> (1)
3: (2) -> (2)
4: (1,1) -> (1,1)
5: (3) -> (3)
6: (2,1) -> (1,2)
7: (4) -> (4)
8: (1,1,1) -> (1,1,1)
9: (2,2) -> (2,1)
10: (3,1) -> (1,3)
11: (5) -> (5)
12: (2,1,1) -> (1,1,2)
13: (6) -> (6)
14: (4,1) -> (1,4)
15: (3,2) -> (2,2)
16: (1,1,1,1) -> (1,1,1,1)
(End)
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
stcinv[q_]:=Total[2^(Accumulate[Reverse[q]])]/2;
stcinv/@Table[Differences[Prepend[primeMS[n], 1]]+1, {n, 100}] (* Gus Wiseman, Dec 23 2022 *)
CROSSREFS
A048793 gives partial sums of reversed standard comps, Heinz number A019565.
A358134 gives partial sums of standard compositions, Heinz number A358170.
Heinz numbers of integer partitions whose augmented differences are all equal.
+10
16
1, 2, 3, 4, 5, 7, 8, 11, 13, 15, 16, 17, 19, 23, 29, 31, 32, 37, 41, 43, 47, 53, 55, 59, 61, 64, 67, 71, 73, 79, 83, 89, 97, 101, 103, 105, 107, 109, 113, 119, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227
COMMENTS
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
The augmented differences aug(y) of an integer partition y of length k are given by aug(y)_i = y_i - y_{i + 1} + 1 if i < k and aug(y)_k = y_k. For example, aug(6,5,5,3,3,3) = (2,1,3,1,1,3).
The enumeration of these partitions by sum is given by A129654.
EXAMPLE
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}
11: {5}
13: {6}
15: {2,3}
16: {1,1,1,1}
17: {7}
19: {8}
23: {9}
29: {10}
31: {11}
32: {1,1,1,1,1}
37: {12}
41: {13}
43: {14}
MATHEMATICA
primeptn[n_]:=If[n==1, {}, Reverse[Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]];
aug[y_]:=Table[If[i<Length[y], y[[i]]-y[[i+1]]+1, y[[i]]], {i, Length[y]}];
Select[Range[100], And@@Table[SameQ@@Differences[aug[primeptn[#]], k], {k, 0, PrimeOmega[#]}]&]
CROSSREFS
Cf. A049988, A056239, A093641, A112798, A129654, A325327, A325328, A325351, A325359, A325366, A325389, A325394, A325395, A325396.
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