cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A351549 Numbers k for which k * gcd(sigma(k), A019565(k)) is equal to sigma(k) * gcd(k, A019565(k)).

Original entry on oeis.org

1, 1456, 15480, 114660, 2244600, 3894768, 25108200, 27052704, 65021040, 112402080, 1973921400
Offset: 1

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Author

Antti Karttunen, Feb 19 2022

Keywords

Comments

Numbers k such that their abundancy index [sigma(k)/k] is equal to A351557(k)/A351556(k).
Question: If the above ratio is neither 1 nor 2, must it then be > 2? Are all even terms abundant?
a(12) > 2281701376 if it exists.

Links

Crossrefs

Cf. A000203, A005101, A007947, A019565, A080398, A351556, A351557, A351558, A351559.
Subsequence of A386424.
Cf. also A351458, A351548.

Programs

  • PARI
    A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
isA351549(n) = { my(s=sigma(n), z=A019565(n)); (n*gcd(s,z))==(s*gcd(n,z)); };

A386425 Odd composites k such that sigma(k) has the same powerful part as k, where sigma is the sum of divisors function.

Original entry on oeis.org

153, 801, 1773, 3725, 4689, 4753, 5013, 6957, 8577, 8725, 9549, 9873, 11493, 13437, 14409, 15381, 18621, 19269, 21213, 21537, 23481, 25101, 26073, 26225, 28989, 29161, 29313, 29961, 32229, 33849, 34173, 36117, 38061, 39033, 40653, 42597, 43893, 47457, 47781, 48725, 48753, 51669, 52317, 54261, 56953, 57177, 57501
Offset: 1

Views

Author

Antti Karttunen, Aug 17 2025

Keywords

Comments

By definition, the sequence contains all odd perfect numbers, and also includes any hypothetical odd triperfect number that is not a multiple of 3 (see A005820 and A347391), and similarly, any odd term of A046060 that is not a multiple of 5, etc. If there are no squares in this sequence (see conjecture in A386424), then the latter categories of numbers certainly do not exist, and this is then a subsequence of A228058.
The first nondeficient term is a(32315) = 81022725. See A386426.

Links

Crossrefs

Intersection of A071904 and A386424.
Nonsquare terms form a subsequence of A228058.
Cf. A000203, A003557, A057521, A386426 (nondeficient terms).
Cf. also A324647, A349749.

Programs

  • Mathematica
    rad[n_] := Times @@ First /@ FactorInteger[n];a057521[n_] := n/Denominator[n/rad[n]^2];Select[Range[9,57501,2],!PrimeQ[#]&&a057521[DivisorSigma[1,#]]==a057521[#]&] (* James C. McMahon, Aug 18 2025 *)
  • PARI
    A057521(n)=my(f=factor(n)); prod(i=1, #f~, if(f[i, 2]>1, f[i, 1]^f[i, 2], 1))
isA386425(n) = ((n>1) && (n%2) && !isprime(n) && (A057521(sigma(n))==A057521(n)));

Formula

{k | k is odd composite and A003557(A000203(k)) = A003557(k)}.

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

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Author

Antti Karttunen, Aug 22 2025

Keywords

Comments

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).

Examples

			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.

Links

Crossrefs

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

Programs

  • PARI
    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)));

Formula

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

A387158 Numbers k such that A173557(k) = A173557(sigma(k)), where A173557(n) is multiplicative with a(p^e) = p-1 and sigma is the sum of divisors function.

Original entry on oeis.org

1, 6, 26, 28, 63, 74, 120, 122, 135, 146, 270, 314, 351, 386, 416, 496, 520, 554, 626, 672, 794, 842, 875, 891, 914, 999, 1080, 1082, 1226, 1232, 1322, 1346, 1404, 1466, 1480, 1514, 1638, 1647, 1750, 1754, 1782, 1859, 1971, 1994, 2186, 2306, 2402, 2426, 2440, 2474, 2642, 2762, 2906, 2920, 3242, 3314, 3506, 3718
Offset: 1

Views

Author

Antti Karttunen, Aug 19 2025

Keywords

Comments

Numbers k for which A173557(k) == A387157(k).

Links

Crossrefs

Cf. A000203, A173557, A387157.
Subsequences: A000396, A387159 (odd terms).
Cf. also A006872, A351446, A386424.

Programs

  • Mathematica
    A387158Q[k_] := #[k] == #[DivisorSigma[1, k]] & [Times @@ (FactorInteger[#][[All, 1]] - 1) &];
Select[Range[10000], A387158Q] (* Paolo Xausa, Aug 20 2025 *)
  • PARI
    A173557(n) = factorback(apply(p -> p-1,factor(n)[,1]));
is_A387158(n) = (A173557(sigma(n))==A173557(n));

A387156 a(n) = A003557(sigma(n)), where A003557(n) is multiplicative with a(p^e) = p^(e-1), and sigma is the sum of divisors function.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 4, 1, 1, 3, 2, 2, 1, 4, 4, 1, 3, 1, 2, 1, 16, 6, 4, 2, 1, 1, 4, 4, 1, 12, 16, 3, 8, 9, 8, 1, 1, 2, 4, 3, 1, 16, 2, 2, 1, 12, 8, 2, 1, 1, 12, 7, 9, 4, 12, 4, 8, 3, 2, 4, 1, 16, 4, 1, 2, 24, 2, 3, 16, 24, 12, 1, 1, 1, 2, 2, 16, 4, 8, 1, 11, 3, 2, 16, 18, 2, 4, 6, 3, 3, 8, 4, 64, 24, 4, 6, 7, 3, 2
Offset: 1

Views

Author

Antti Karttunen, Aug 19 2025

Keywords

Links

Crossrefs

Cf. A000203, A003557, A062401, A080398, A387157.
Cf. also A386424, A386425.

Programs

  • Mathematica
    A387156[n_] := # / Times @@ FactorInteger[#][[All, 1]] & [DivisorSigma[1, n]];
Array[A387156, 100] (* Paolo Xausa, Aug 20 2025 *)
  • PARI
    A387156(n) = { my(s=sigma(n)); s/factorback(factor(s)[,1]); };

Formula

a(n) = A000203(n) / A080398(n).
a(n) = A062401(n) / A387157(n).
Showing 1-5 of 5 results.

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