Displaying 1-10 of 21 results found.
0, 0, 0, 1, 0, 1, 0, 1, 6, 0, 0, 1, 0, 1, 8, 9, 0, 1, 0, 1, 16, 1, 0, 1, 0, 1, 10, 1, 0, 1, 0, 1, 0, 1, 20, 9, 0, 1, 66, 1, 0, 1, 0, 1, 6, 1, 0, 1, 0, 0, 2, 1, 0, 1, 36, 1, 258, 1, 0, 1, 0, 1, 6, 41, 0, 1, 0, 1, 0, 1, 0, 17, 0, 1, 16, 1, 32, 1, 0, 1, 10, 1, 0, 1, 132, 1, 1026, 1, 0, 33, 72, 1, 0, 1, 256, 25, 0, 0, 66, 17, 0, 1, 0, 1, 34
PROG
(PARI)
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1, 1]==2), f[1, 2] = 0); for (i=1, #f~, f[i, 1] = precprime(f[i, 1]-1)); factorback(f)};
A318458(n) = bitand(n, sigma(n)-n);
Numbers k such that A318458(k) (bitwise-AND of k and sigma(k)-k) is equal to k.
+20
7
6, 20, 28, 36, 66, 72, 88, 100, 104, 114, 132, 150, 240, 258, 264, 272, 280, 304, 354, 368, 392, 402, 464, 496, 498, 516, 550, 552, 642, 644, 680, 708, 748, 770, 774, 784, 786, 834, 836, 840, 860, 978, 1026, 1032, 1040, 1044, 1056, 1062, 1064, 1068, 1074, 1092, 1104, 1120, 1184, 1232, 1266, 1312, 1362, 1376, 1410, 1504
COMMENTS
Positions of zeros in A324648. Fixed points of A318458, also positions of the records in the latter. The odd terms are: 7425, 76545, 92565, ... ( A324897).
PROG
(PARI) for(n=1, oo, if(bitand(n, sigma(n)-n)==n, print1(n, ", ")));
Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [ A009194(n), A318458(n)] for all other numbers, except f(1) = -1.
+20
6
1, 2, 3, 2, 3, 4, 3, 2, 2, 5, 3, 6, 3, 7, 8, 2, 3, 9, 3, 10, 3, 11, 3, 12, 2, 13, 14, 15, 3, 16, 3, 2, 17, 18, 3, 19, 3, 11, 3, 20, 3, 21, 3, 22, 23, 7, 3, 6, 2, 24, 25, 26, 3, 27, 28, 29, 28, 30, 3, 31, 3, 32, 33, 2, 3, 34, 3, 18, 17, 35, 3, 36, 3, 5, 3, 37, 3, 38, 3, 39, 2, 18, 3, 40, 41, 11, 17, 42, 3, 43, 44, 45, 3, 46, 47, 12, 3, 48, 23, 49, 3, 50, 3
PROG
(PARI)
up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om, invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om, invec[i], i); outvec[i] = u; u++ )); outvec; };
A318458(n) = bitand(n, sigma(n)-n);
v324389 = rgs_transform(vector(up_to, n, Aux324389(n)));
Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [ A033879(n), A318458(n)] for all other numbers, except f(1) = -1.
+20
5
1, 2, 3, 2, 4, 5, 6, 2, 7, 8, 9, 10, 11, 12, 13, 2, 14, 15, 16, 17, 9, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 2, 16, 28, 19, 29, 30, 31, 19, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 2, 39, 56, 57, 58, 35, 59, 60, 61, 62, 63, 64, 65, 51, 66, 67, 68, 41, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 51, 80
FORMULA
a(2^n) = 2 for all n >= 1.
PROG
(PARI)
up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om, invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om, invec[i], i); outvec[i] = u; u++ )); outvec; };
A318458(n) = bitand(n, sigma(n)-n);
v324530 = rgs_transform(vector(up_to, n, Aux324530(n)));
Odd numbers k such that sigma(k) is congruent to 2 modulo 4 and k = A318458(k), where A318458(k) is bitwise-AND of k and sigma(k)-k.
+20
5
236925, 3847725, 51122925, 69468525, 151141725, 154669725, 269748225, 344211525, 415565325, 445817925, 551569725, 1111904325, 1112565825, 1113756525, 1175717025, 1400045625, 1631666925, 1695170925, 1820873925, 1915847325, 1946981925, 2179080225, 2321121825, 2453690925, 2460041325, 2491740225, 3223500525, 3493517445, 3775103325
COMMENTS
If this sequence has no common terms with A324647, or no terms common with A324727, then there are no odd perfect numbers. The first 29 terms factored:
236925 = 3^6 * 5^2 * 13,
3847725 = 3^2 * 5^2 * 7^2 * 349,
51122925 = 3^2 * 5^2 * 7^2 * 4637,
69468525 = 3^2 * 5^2 * 7^2 * 6301,
151141725 = 3^2 * 5^2 * 7^2 * 13709,
154669725 = 3^2 * 5^2 * 7^2 * 14029,
269748225 = 3^6 * 5^2 * 19^2 * 41,
344211525 = 3^4 * 5^2 * 7^2 * 3469,
415565325 = 3^2 * 5^2 * 7^2 * 37693,
445817925 = 3^4 * 5^2 * 7^2 * 4493,
551569725 = 3^2 * 5^2 * 7^4 * 1021,
1111904325 = 3^2 * 5^2 * 7^2 * 100853,
1112565825 = 3^2 * 5^2 * 7^2 * 100913,
1113756525 = 3^2 * 5^2 * 7^2 * 101021,
1175717025 = 3^4 * 5^2 * 7^2 * 17^2 * 41,
1400045625 = 3^2 * 5^4 * 11^4 * 17,
1631666925 = 3^2 * 5^2 * 7^2 * 147997,
1695170925 = 3^2 * 5^2 * 7^2 * 153757,
1820873925 = 3^4 * 5^2 * 13 * 263^2, [Here the unitary prime is not the largest]
1915847325 = 3^2 * 5^2 * 7^2 * 173773,
1946981925 = 3^2 * 5^2 * 7^2 * 176597,
2179080225 = 3^4 * 5^2 * 7^2 * 21961,
2321121825 = 3^4 * 5^2 * 11^2 * 9473,
2453690925 = 3^2 * 5^2 * 7^2 * 222557,
2460041325 = 3^2 * 5^2 * 7^2 * 223133,
2491740225 = 3^6 * 5^2 * 13^2 * 809,
3223500525 = 3^2 * 5^2 * 7^2 * 292381,
3493517445 = 3^6 * 5^1 * 11^2 * 89^2, [Here the unitary prime is not the largest]
3775103325 = 3^2 * 5^2 * 7^2 * 342413.
Subsequence of A228058 provided this sequence does not contain any prime powers. - Antti Karttunen, Jun 17 2019 Sequence contains no prime powers up to 10^20. I believe any prime powers must be of the form (4k+1)^(4e+1), in which case I have verified this up to 10^50. - Charles R Greathouse IV, Dec 08 2021
MATHEMATICA
Select[Range[10^5, 10^8, 2], And[Mod[#2, 4] == 2, BitAnd[#1, #2 - #1] == #1] & @@ {#, DivisorSigma[1, #]} &] (* Michael De Vlieger, Jun 22 2019 *)
PROG
(PARI) for(n=1, oo, if((n%2)&&2==((t=sigma(n))%4)&&(bitand(n, t-n)==n), print1(n, ", ")));
Lexicographically earliest sequence such that for all i, j >= 1, a(i) = a(j) => f(i) = f(j), where f(n) = [ A278222(n), A318458(n)] for all other numbers, except f(1) = 0.
+20
4
1, 2, 3, 2, 4, 5, 6, 2, 7, 8, 9, 10, 9, 11, 12, 2, 4, 13, 9, 14, 15, 16, 17, 10, 18, 19, 20, 21, 17, 22, 23, 2, 4, 7, 9, 24, 15, 16, 17, 25, 15, 26, 27, 28, 29, 30, 31, 10, 18, 32, 33, 34, 27, 35, 36, 37, 38, 39, 40, 41, 31, 42, 43, 2, 4, 44, 9, 7, 15, 45, 17, 46, 15, 47, 27, 48, 27, 49, 31, 50, 51, 51, 27, 52, 53, 54, 55, 56, 27, 57, 58, 59, 55, 60, 61, 10, 9, 48
PROG
(PARI)
up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om, invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om, invec[i], i); outvec[i] = u; u++ )); outvec; };
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ Modified from code of M. F. Hasler
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ This function from Charles R Greathouse IV, Aug 17 2011
A318458(n) = bitand(n, sigma(n)-n);
v324531 = rgs_transform(vector(up_to, n, Aux324531(n)));
a(n) = n - A318458(n), where A318458(n) is bitwise-AND of n and the sum of proper divisors of n (sigma(n)-n).
+20
4
1, 2, 2, 4, 4, 0, 6, 8, 9, 2, 10, 12, 12, 4, 6, 16, 16, 2, 18, 0, 20, 16, 22, 24, 25, 10, 18, 0, 28, 20, 30, 32, 32, 34, 34, 0, 36, 32, 38, 8, 40, 8, 42, 4, 12, 36, 46, 48, 49, 16, 34, 16, 52, 52, 38, 56, 40, 26, 58, 16, 60, 28, 22, 64, 64, 0, 66, 68, 68, 4, 70, 0, 72, 66, 74, 12, 76, 4, 78, 16, 81, 82, 82, 80, 64, 80, 86, 0, 88
MATHEMATICA
Array[# - BitAnd[#, DivisorSigma[1, #] - #] &, 100] (* Paolo Xausa, Mar 12 2024 *)
PROG
(PARI)
A318458(n) = bitand(n, sigma(n)-n);
(PARI) A324648(n) = (n-bitand(n, sigma(n)-n));
Odd numbers k such that A318458(k) (bitwise-AND of k and sigma(k)-k) is equal to k.
+20
4
7425, 76545, 92565, 236925, 831105, 954765, 1401345, 2011905, 2048445, 2129985, 2253825, 2445345, 2621745, 2974725, 3283245, 3847725, 5709825, 6447105, 8422785, 8503425, 8945685, 10781505, 12488385, 13470345, 14322945, 15213825, 15340545, 19470465, 19502145, 20075265, 22749825, 25740225, 25756605, 26215245, 27009045
COMMENTS
If this sequence has no common terms with A324647, or no terms common with A324727, then there are no odd perfect numbers. The first 16 terms factored:
7425 = 3^3 * 5^2 * 11,
76545 = 3^7 * 5 * 7,
92565 = 3^2 * 5 * 11^2 * 17,
236925 = 3^6 * 5^2 * 13,
831105 = 3^2 * 5 * 11 * 23 * 73,
954765 = 3^2 * 5 * 7^2 * 433,
1401345 = 3^2 * 5 * 11 * 19 * 149,
2011905 = 3^3 * 5 * 7 * 2129,
2048445 = 3^2 * 5 * 7^2 * 929,
2129985 = 3^2 * 5 * 11 * 13 * 331,
2253825 = 3^5 * 5^2 * 7 * 53,
2445345 = 3^2 * 5 * 7^2 * 1109,
2621745 = 3^2 * 5 * 7^2 * 29 * 41,
2974725 = 3^4 * 5^2 * 13 * 113,
3283245 = 3^2 * 5 * 7^2 * 1489,
3847725 = 3^2 * 5^2 * 7^2 * 349.
PROG
(PARI) isok(k) = (k%2) && (bitand(k, sigma(k)-k) == k); \\ Michel Marcus, Jul 18 2021
Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [ A000120(n), A318458(n)] for all other numbers, except f(1) = 0.
+20
3
1, 2, 3, 2, 3, 4, 5, 2, 6, 7, 5, 6, 5, 8, 9, 2, 3, 10, 5, 11, 5, 12, 13, 6, 14, 15, 9, 16, 13, 17, 18, 2, 3, 6, 5, 19, 5, 12, 13, 20, 5, 21, 13, 22, 23, 17, 18, 6, 14, 21, 24, 25, 13, 26, 27, 14, 24, 28, 18, 29, 18, 30, 31, 2, 3, 32, 5, 6, 5, 33, 13, 34, 5, 35, 13, 36, 13, 37, 18, 38, 14, 14, 13, 39, 40, 41, 18, 42, 13, 43, 27, 44, 18, 45, 46, 6, 5, 36, 23, 47, 13
PROG
(PARI)
up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om, invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om, invec[i], i); outvec[i] = u; u++ )); outvec; };
A318458(n) = bitand(n, sigma(n)-n);
Aux324532(n) = if(1==n, 0, [hammingweight(n), A318458(n)]); v324532 = rgs_transform(vector(up_to, n, Aux324532(n)));
Lexicographically earliest infinite sequence such that a(i) = a(j) => A318458(i) = A318458(j) and A336158(i) = A336158(j), for all i, j >= 1.
+20
3
1, 1, 2, 1, 2, 3, 2, 1, 4, 5, 2, 6, 2, 7, 8, 1, 2, 9, 2, 10, 11, 3, 2, 6, 4, 12, 13, 14, 2, 15, 2, 1, 11, 6, 11, 16, 2, 3, 11, 17, 2, 18, 2, 19, 20, 7, 2, 6, 4, 21, 22, 23, 2, 24, 22, 6, 22, 17, 2, 25, 2, 26, 27, 1, 11, 28, 2, 6, 11, 28, 2, 29, 2, 5, 30, 31, 11, 32, 2, 31, 33, 6, 2, 34, 35, 3, 11, 36, 2, 37, 22, 38, 11, 39, 40, 6, 2, 41, 20, 42, 2, 43, 2, 44, 45
COMMENTS
Restricted growth sequence transform of the ordered pair [ A318458(n), A336158(n)]. For all i, j:
PROG
(PARI)
up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om, invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om, invec[i], i); outvec[i] = u; u++ )); outvec; };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ From A046523
A318458(n) = bitand(n, sigma(n)-n);
v336157 = rgs_transform(vector(up_to, n, Aux336157(n)));
Search completed in 0.014 seconds
| 