A351555 -id:A351555 - cp's OEIS frontend

A351551 Numbers k such that the largest unitary divisor of sigma(k) that is coprime with A003961(k) is also a unitary divisor of k.

1, 2, 10, 34, 106, 120, 216, 260, 340, 408, 440, 580, 672, 696, 820, 1060, 1272, 1666, 1780, 1940, 2136, 2340, 2464, 3320, 3576, 3960, 4280, 4536, 5280, 5380, 5860, 6456, 6960, 7520, 8746, 8840, 9120, 9632, 10040, 10776, 12528, 12640, 13464, 14560, 16180, 16660, 17400, 17620, 19040, 19416, 19992, 21320, 22176, 22968

Offset: 1

Antti Karttunen, Feb 16 2022

nonn

Numbers k for which A351546(k) is a unitary divisor of k.
The condition guarantees that A351555(k) = 0, therefore this is a subsequence of A351554.
The condition is also a necessary condition for A349745, therefore it is a subsequence of this sequence.
All six known 3-perfect numbers (A005820) are included in this sequence.
All 65 known 5-multiperfects (A046060) are included in this sequence.
Not all multiperfects (A007691) are present (only 587 of the first 1600 are), but all 23 known terms of A323653 are terms, while none of the (even) terms of A046061 or A336702 are.

			For n = 672 = 2^5 * 3^1 * 7^1, and the largest unitary divisor of the sigma(672) [= 2^5 * 3^2 * 7^1] coprime with A003961(672) [= 13365 = 3^5 * 5^1 * 11^1] is 2^5 * 7^1 = 224, therefore A351546(672) is a unitary divisor of 672, and 672 is included in this sequence.

Cf. A000203, A000396, A003961, A007691, A046061, A065997, A336702, A351546, A351555, A353633 (characteristic function).
Subsequence of A351552 and of A351554.
Cf. A349745, A351550 (subsequences), A005820, A046060, A323653 (very likely subsequences).

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A351546(n) = { my(f=factor(sigma(n)),u=A003961(n)); prod(k=1,#f~,f[k,1]^((0!=(u%f[k,1]))*f[k,2])); };
isA351551(n) =  { my(u=A351546(n)); (!(n%u) && 1==gcd(u,n/u)); };

A351554 Numbers k such that there are no odd prime factors p of sigma(k) such that p does not divide A003961(k) and the valuation(k, p) is different from valuation(sigma(k), p), where A003961 is fully multiplicative with a(p) = nextprime(p), and sigma is the sum of divisors function.

Original entry on oeis.org

1, 2, 3, 6, 7, 10, 14, 15, 20, 21, 22, 24, 27, 28, 30, 31, 33, 34, 40, 42, 46, 54, 57, 60, 62, 66, 69, 70, 84, 87, 91, 93, 94, 102, 105, 106, 110, 114, 120, 127, 130, 138, 140, 141, 142, 154, 160, 168, 170, 174, 177, 182, 186, 189, 190, 195, 198, 210, 214, 216, 217, 220, 224, 230, 231, 237, 238, 254, 260, 264, 270, 273

Offset: 1

Antti Karttunen, Feb 16 2022

Numbers k for which A351555(k) = 0. This is a necessary condition for the terms of A349169 and of A349745, therefore they are subsequences of this sequence.
All six known 3-perfect numbers (A005820) are included in this sequence.
All 65 known 5-multiperfects (A046060) are included in this sequence.
Moreover, all multiperfect numbers (A007691) seem to be in this sequence.
From Antti Karttunen, Aug 27 2025: (Start)
Multiperfect number m is included in this sequence only if its abundancy sigma(m)/m has only such odd prime factors p that prevprime(p) [A151799] divides m for each p.  E.g., all 65 known 5-multiperfects are multiples of 3, and all known terms of A005820 and A046061 are even.
This sequence contains natural numbers k such that the odd primes in the prime factorization of sigma(k) have the same valuation there as in k, except that the primes in A003961(k) [or equally in A003961(A007947(k))] stand for "don't care primes", that are "masked off" from the comparison.
(End)

Cf. A000203, A003961, A151799, A351546.
Positions of zeros in A351555.
Subsequences: A000396, A351553 (even terms), A386430 (odd terms), A351551, A349169, A349745, A387160 (terms of the form prime * m^2), also these, at least all the currently (Feb 2022) known terms: A005820, A007691, A046060.

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A351555(n) = { my(s=sigma(n),f=factor(s),u=A003961(n)); sum(k=1,#f~,if((f[k,1]%2) && 0!=(u%f[k,1]), (valuation(n,f[k,1])!=f[k,2]), 0)); };
isA351554(n) = (0==A351555(n));

isA351554(n) = { my(sh=A351546(n),f=factor(sh)); for(i=1,#f~, if((f[i,1]%2)&&valuation(n,f[i,1])!=f[i,2],return(0))); (1); }; \\ Uses also program given in A351546.

Definition corrected by Antti Karttunen, Aug 22 2025

A351538 Numbers k such that both k and sigma(k) are congruent to 2 modulo 4 and the 3-adic valuation of sigma(k) is exactly 1.

Original entry on oeis.org

26, 74, 122, 146, 194, 218, 234, 314, 362, 386, 458, 482, 554, 626, 650, 666, 674, 698, 746, 794, 818, 842, 866, 914, 1082, 1098, 1154, 1202, 1226, 1314, 1322, 1346, 1418, 1466, 1514, 1538, 1658, 1706, 1746, 1754, 1850, 1874, 1962, 1994, 2018, 2042, 2066, 2106, 2138, 2186, 2234, 2258, 2306, 2402, 2426, 2474, 2498

Offset: 1

Antti Karttunen, Feb 14 2022

nonn

All the terms of the form 4u+2 in A349745 (if they exist) are found in this sequence. As A351537 is the intersection of A191218 and A329963, and the latter has asymptotic density zero, so has this sequence also. It is conjectured that A351555(a(n)) is nonzero for all n, which would imply that the intersection with A349745 is empty. - Antti Karttunen, Feb 19 2022

Antti Karttunen, Table of n, a(n) for n = 1..25000
Index entries for sequences related to sigma(n)

Cf. A191218, A329963, A349745, A351537.

Probably a subsequence of A351543. (See also A351550, A351555).

isA351538(n) = if(!(2==(n%4)),0, my(s=sigma(n)); (2 == (s%4)) && (1==valuation(s,3)));

a(n) = 2 * A351537(n).

A351543 Even numbers k such that there is an odd prime p that divides sigma(k), but valuation(k, p) differs from valuation(sigma(k), p), and p does not divide A003961(k), where A003961 is fully multiplicative with a(p) = nextprime(p), and sigma is the sum of divisors function.

Original entry on oeis.org

4, 8, 12, 16, 18, 26, 32, 36, 38, 44, 48, 50, 52, 56, 58, 64, 68, 72, 74, 76, 78, 80, 82, 86, 88, 90, 92, 96, 98, 100, 104, 108, 112, 116, 118, 122, 124, 126, 128, 132, 134, 136, 144, 146, 148, 150, 152, 156, 158, 162, 164, 166, 172, 176, 178, 180, 184, 188, 192, 194, 196, 200, 202, 204, 206, 208, 212, 218, 222, 226

Offset: 1

Antti Karttunen, Feb 16 2022

nonn

Even numbers k such that sigma(k) has an odd prime factor prime(i), but prime(i-1) is not a factor of k, and A286561(k, prime(i)) <> A286561(sigma(k), prime(i)). This differs from the definition of A351542 in that prime(i) is not here required to be a factor of k itself. The condition implies also that if there is any such odd prime factor prime(i) of sigma(k), it must be >= 5.
Even numbers k for which A351555(k) > 0.
Question: Is A351538 subsequence of this sequence?

			12 = 2^2 * 3 is present as sigma(12) = 28 = 2^2 * 7, whose prime factorization contains an odd prime 7 such that neither it nor the immediately previous prime, which is 5, divide 12 itself.
196 = 2^2 * 7^2 is present as sigma(196) = 399 = 3^1 * 7^1 * 19^1, which thus has a shared prime factor 7 with 196, but occurring with smaller exponent, and with no prime 5 (which is the previous prime before 7) present in the prime factorization of 196.
364 = 2^2 * 7^1 * 13^1 is present as sigma(364) = 784 = 2^4 * 7^2, which thus has a shared prime factor 7 with 364, but occurring with larger exponent, and with no prime 5 (which is the previous prime before 7) present in the prime factorization of 364.

Cf. A000203, A003961, A286561, A351555.
Subsequences: A351541, A351542, and also conjecturally A351538.
Cf. A351553 (complement among even numbers).
No common terms with A349745.

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A351555(n) = { my(s=sigma(n),f=factor(s),u=A003961(n)); sum(k=1,#f~,if((f[k,1]%2) && 0!=(u%f[k,1]), (valuation(n,f[k,1])!=f[k,2]), 0)); };
isA351543(n) = (!(n%2) && A351555(n)>0);

A386430 Odd numbers k such that there are no prime factors p of sigma(k) such that p does not divide A003961(k) and the valuation(k, p) is different from valuation(sigma(k), p), where A003961 is fully multiplicative with a(p) = nextprime(p), and sigma is the sum of divisors function.

Original entry on oeis.org

1, 3, 7, 15, 21, 27, 31, 33, 57, 69, 87, 91, 93, 105, 127, 141, 177, 189, 195, 217, 231, 237, 273, 285, 301, 381, 399, 447, 465, 483, 495, 513, 567, 573, 597, 609, 627, 651, 717, 775, 819, 837, 861, 889, 903, 987, 1023, 1029, 1149, 1185, 1239, 1311, 1365, 1419, 1431, 1437, 1455, 1497, 1561, 1653, 1659, 1687, 1743

Offset: 1

Antti Karttunen, Aug 22 2025

Conjecture: After the initial 1, and apart from any hypothetical odd perfect numbers, all other terms are in A248150, i.e., sigma(k) == 0 (mod 4). This would imply (with the same caveat), that this sequence has no common terms with A228058 and no squares larger than one. This is true at least for the first 709203 terms (terms in range [1..2^34]).

Terms k such that A162642(k) = 1 are rare: 3, 7, 27, 31, 127, 567, 775, 8191, 27783, 131071, 524287, 2147483647, ... (odd terms of A387160).

			a(386548) = 5919068925 = 3^4 * 5^2 * 7^2 * 11^2 * 17 * 29. sigma(5919068925) = 15355618740 = 2^2 * 3^4 * 5 * 7 * 11^2 * 19^2 * 31. The "don't care primes" is given by A003961(A007947(5919068925))) = 2947945 = 5*7*11*13*19*31, thus only odd prime factor that matters here is 3, which in case has the same exponent (4) in both n = 5919068925 and sigma(n). In a way, this number is very close to satisfying Euler's criterion for odd perfect numbers (A228058), except that it has two unitary prime factors of the form 4k+1, instead of just one, apart from the square factor. Both n/17 and n/29 are in A228058.

Odd terms of A351554.
Cf. A000203, A003961, A056169, A162642, A228058, A248150, A351555, A386424, A387160.
Cf. A349169 (subsequence).

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A351555(n) = { my(s=sigma(n),f=factor(s),u=A003961(n)); sum(k=1,#f~,if((f[k,1]%2) && 0!=(u%f[k,1]), (valuation(n,f[k,1])!=f[k,2]), 0)); };
isA386430(n) = ((n%2) && (0==A351555(n)));

{k | k odd, A351555(k) = 0}.

A351553 Even numbers k such that there are no odd prime factors p of sigma(k) such that p does not divide A003961(k) and the valuation(k, p) is different from valuation(sigma(k), p), where A003961 is fully multiplicative with a(p) = nextprime(p), and sigma is the sum of divisors function.

Original entry on oeis.org

2, 6, 10, 14, 20, 22, 24, 28, 30, 34, 40, 42, 46, 54, 60, 62, 66, 70, 84, 94, 102, 106, 110, 114, 120, 130, 138, 140, 142, 154, 160, 168, 170, 174, 182, 186, 190, 198, 210, 214, 216, 220, 224, 230, 238, 254, 260, 264, 270, 280, 282, 290, 308, 310, 318, 322, 330, 340, 354, 374, 378, 380, 382, 390, 408, 410, 420, 426

Offset: 1

Antti Karttunen, Feb 16 2022

Even numbers k for which A351555(k) = 0.

Cf. A000203, A005820, A003961, A046060, A351552.
Even terms in A351554, positions of zeros at even indices in A351555.
Cf. A351543 (complement among even numbers), A386430.

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; A351555(n) = { my(s=sigma(n),f=factor(s),u=A003961(n)); sum(k=1,#f~,if((f[k,1]%2) && 0!=(u%f[k,1]), (valuation(n,f[k,1])!=f[k,2]), 0)); };
isA351553(n) = (!(n%2) && 0==A351555(n));

Definition corrected by Antti Karttunen, Aug 27 2025

A387160 Terms of the form prime * m^2 in A351554.

Original entry on oeis.org

2, 3, 7, 20, 27, 28, 31, 127, 496, 567, 775, 2268, 3100, 8128, 8191, 27783, 131071, 403172, 524287, 3628548, 17389708, 32656932, 33550336, 127926848, 1087307452, 1248461136, 1408566348, 2147483647, 7802882100, 8589869056, 9785767068, 10362074688, 31211528400, 88071903612

Offset: 1

Antti Karttunen, Aug 24 2025

Conjecture: This sequence has no common terms with A228058. See comments in A386430.

Intersection of A229125 and A351554.
Cf. A162642, A228058, A351555, A386430.
Subsequences: A000396\{6}, A000668.

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A351555(n) = { my(s=sigma(n),f=factor(s),u=A003961(n)); sum(k=1,#f~,if((f[k,1]%2) && 0!=(u%f[k,1]), (valuation(n,f[k,1])!=f[k,2]), 0)); };
isA387160(n) = (isprime(core(n)) && (0==A351555(n)));

{k | A162642(k) = 1 and A351555(k) = 0}.

Terms a(29)-a(34) from Giovanni Resta, Aug 25 2025

cp's OEIS Frontend

A351551 Numbers k such that the largest unitary divisor of sigma(k) that is coprime with A003961(k) is also a unitary divisor of k.

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A351554 Numbers k such that there are no odd prime factors p of sigma(k) such that p does not divide A003961(k) and the valuation(k, p) is different from valuation(sigma(k), p), where A003961 is fully multiplicative with a(p) = nextprime(p), and sigma is the sum of divisors function.

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A351538 Numbers k such that both k and sigma(k) are congruent to 2 modulo 4 and the 3-adic valuation of sigma(k) is exactly 1.

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A351543 Even numbers k such that there is an odd prime p that divides sigma(k), but valuation(k, p) differs from valuation(sigma(k), p), and p does not divide A003961(k), where A003961 is fully multiplicative with a(p) = nextprime(p), and sigma is the sum of divisors function.

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A386430 Odd numbers k such that there are no prime factors p of sigma(k) such that p does not divide A003961(k) and the valuation(k, p) is different from valuation(sigma(k), p), where A003961 is fully multiplicative with a(p) = nextprime(p), and sigma is the sum of divisors function.

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A351553 Even numbers k such that there are no odd prime factors p of sigma(k) such that p does not divide A003961(k) and the valuation(k, p) is different from valuation(sigma(k), p), where A003961 is fully multiplicative with a(p) = nextprime(p), and sigma is the sum of divisors function.

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A387160 Terms of the form prime * m^2 in A351554.

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