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 1-9 of 9 results.

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

Views

Author

Antti Karttunen, Aug 24 2025

Keywords

Comments

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

Links

Crossrefs

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

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)); };
isA387160(n) = (isprime(core(n)) && (0==A351555(n)));

Formula

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

Extensions

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

A323653 Multiperfect numbers m such that sigma(m) is also multiperfect.

Original entry on oeis.org

1, 459818240, 51001180160, 13188979363639752997731839211623940096, 5157152737616023231698245840143799191339008, 54530444405217553992377326508106948362108928, 133821156044600922812153118065015159487725568, 4989680372093758991515359988337845750507257510078971904
Offset: 1

Views

Author

Jaroslav Krizek, Jan 21 2019

Keywords

Comments

Multiperfect numbers m such that sigma(m) divides sigma(sigma(m)).
Also k-multiperfect numbers m such that k*m is also multiperfect.
Corresponding values of numbers k(n) = sigma(a(n)) / a(n): 1, 3, 3, 5, 5, 5, 5, 5, ...
Corresponding values of numbers h(n) = sigma(k(n) * a(n)) / (k(n) * a(n)): 1, 4, 4, 6, 6, 6, 6, 6, ...
Number of k-multiperfect numbers m such that sigma(n) is also multiperfect for k = 3..6: 2, 0, 20, 0.
From Antti Karttunen, Mar 20 2021, Feb 18 2022: (Start)
Conjecture 1 (a): This sequence consists of those m for which sigma(m)/m is an integer (thus a term of A007691), and coprime with m. Or expressed in a slightly weaker form (b): {1} followed by those m for which sigma(m)/m is an integer, but not a divisor of m. In a slightly stronger form (c): For m > 1, sigma(m)/m is always the least prime not dividing m. This would imply both (a) and (b) forms.
Conjecture 2: This sequence is finite.
Conjecture 3: This sequence is the intersection of A007691 and A351458.
Conjecture 4: This is a subsequence of A349745, thus also of A351551 and of A351554.
Note that if there existed an odd perfect number x that were not a multiple of 3, then both x and 2*x would be terms in this sequence, as then we would have: sigma(x)/x = 2, sigma(2*x)/(2*x) = 3, sigma(6*x)/(6*x) = 4. See also the diagram in A347392 and A353365.
(End)
From Antti Karttunen, May 16 2022: (Start)
Apparently for all n > 1, A336546(a(n)) = 0. [At least for n=2..23], while A353633(a(n)) = 1, for n=1..23.
The terms a(1) .. a(23) are only cases present among the 5721 known and claimed multiperfect numbers with abundancy <> 2, as published 03 January 2022 under Flammenkamp's site, that satisfy the condition for inclusion in this sequence.
They are also the only 23 cases among that data such that gcd(n, sigma(n)/n) = 1, or in other words, for which the n and its abundancy are relatively prime, with abundancy in all cases being the least prime that does not divide n, A053669(n), which is a sufficient condition for inclusion in A351458.
(End)

Examples

			3-multiperfect number 459818240 is a term because number 3*459818240 = 1379454720 is a 4-multiperfect number.

Links

Crossrefs

Cf. A000396, A005820, A027687, A046060, A046061, A053669, A054030, A145551, A323652, A336546, A342659, A347392, A349745, A351458, A351551, A351554, A353365.
Intersection of any two of these three sequences: A007691, A019278, A066961.

Programs

  • Magma
    [n: n in [1..10^6] | SumOfDivisors(n) mod n eq 0 and SumOfDivisors(SumOfDivisors(n)) mod SumOfDivisors(n) eq 0];
  • PARI
    ismulti(n) = (sigma(n) % n) == 0;
isok(n) = ismulti(n) && ismulti(sigma(n)); \\ Michel Marcus, Jan 26 2019

A349162 a(n) = sigma(n) / gcd(sigma(n), A003961(n)), where A003961 shifts the prime factorization of n one step towards larger primes, and sigma is the sum of divisors function.

Original entry on oeis.org

1, 1, 4, 7, 6, 4, 8, 5, 13, 6, 12, 28, 14, 8, 24, 31, 18, 13, 20, 2, 32, 12, 24, 4, 31, 14, 8, 56, 30, 24, 32, 7, 48, 18, 48, 91, 38, 20, 56, 10, 42, 32, 44, 28, 78, 24, 48, 124, 57, 31, 72, 98, 54, 8, 72, 40, 16, 30, 60, 8, 62, 32, 104, 127, 12, 48, 68, 14, 96, 48, 72, 13, 74, 38, 124, 140, 96, 56, 80, 62, 121, 42
Offset: 1

Views

Author

Antti Karttunen, Nov 09 2021

Keywords

Comments

Denominator of ratio A003961(n) / A000203(n).
Small values are rare, but are not limited to the beginning. For example in range 1 .. 2^25, a(n) = 4 at n = 3, 6, 24, 792, 2720, 122944, 31307472.
Question: Would it be possible to prove that a(n) > 1 for all n > 2?
Obviously, 1's may occur only on squares & twice squares (A028982). See also comments in A350072. - Antti Karttunen, Feb 16 2022

Links

Crossrefs

Cf. A000203, A003961, A028982 (positions of odd terms), A319630, A336702, A342671, A348992 (the odd part), A348993, A349161 (numerators), A349163, A349164, A349627, A349628, A350072 [= a(n^2)].
Cf. also A349745, A351551, A351554.

Programs

  • Mathematica
    Array[#1/GCD[##] & @@ {DivisorSigma[1, #], If[# == 1, 1, Times @@ Map[NextPrime[#1]^#2 & @@ # &, FactorInteger[#]]]} &, 82] (* Michael De Vlieger, Nov 11 2021 *)
  • PARI
    A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A349162(n) = { my(s=sigma(n)); (s/gcd(s,A003961(n))); };

Formula

a(n) = A000203(n) / A342671(n) = A000203(n) / gcd(A000203(n), A003961(n)).

A349745 Numbers k for which k * gcd(sigma(k), A003961(k)) is equal to sigma(k) * gcd(k, A003961(k)), where A003961 shifts the prime factorization one step towards larger primes, and sigma is the sum of divisors function.

Original entry on oeis.org

1, 120, 216, 672, 2464, 22176, 228480, 523776, 640640, 837760, 5581440, 5765760, 7539840, 12999168, 19603584, 33860736, 38342304, 71344000, 95472000, 102136320, 197308800, 220093440, 345080736, 459818240, 807009280, 975576960, 1476304896, 1510831360, 1773584640
Offset: 1

Views

Author

Antti Karttunen, Nov 29 2021

Keywords

Comments

Numbers k for which k * A342671(k) = A000203(k) * A322361(k).
Numbers k such that gcd(A064987(k), A191002(k)) = gcd(A064987(k), A341529(k)).
Obviously, all odd terms in this sequence must be squares.
All the terms k of A005820 that satisfy A007949(k) < A007814(k) [i.e., whose 3-adic valuation is strictly less than their 2-adic valuation] are also terms of this sequence. Incidentally, the first six known terms of A005820 satisfy this condition, while on the other hand, any hypothetical odd 3-perfect number would be excluded from this sequence. Also, as a corollary, any hypothetical 3-perfect numbers of the form 4u+2 must not be multiples of 3 if they are to appear here. Similarly for any k which occurs in A349169, for 2*k to occur in this sequence, it shouldn't be a multiple of 3 and k should also be a term of A191218. See question 2 and its partial answer in A349169.
From Antti Karttunen, Feb 13-20 2022: (Start)
Question: Are all terms/2 (A351548) abundant, from n > 1 onward?
Note that of the 65 known 5-multiperfect numbers, all others except these three 1245087725796543283200, 1940351499647188992000, 4010059765937523916800 are also included in this sequence. The three exceptions are distinguished by the fact that their 3 and 5-adic valuations are equal. In 62 others the former is larger.
If k satisfying the condition were of the form 4u+2, then it should be one of the terms of A191218 doubled as only then both k and sigma(k) are of the form 4u+2, with equal 2-adic valuations for both. More precisely, one of the terms of A351538.
(End)

Links

Crossrefs

Cf. A000203, A003961, A005101, A005820, A007814, A007949, A046060, A064987, A191002, A191218, A248150, A322361, A341529, A342671, A347870, A351534, A351536, A351538.
Cf. also A349169, A349746, A351458, A351549 for other variants.
Subsequence of A351554 and also of its subsequence A351551.
Cf. A351459 (subsequence, intersection with A351458), A351548 (terms halved).

Programs

  • Mathematica
    f1[p_, e_] := (p^(e + 1) - 1)/(p - 1); f2[p_, e_] := NextPrime[p]^e; q[1] = True; q[n_] := n * GCD[(s = Times @@ f1 @@@ (f = FactorInteger[n])), (r = Times @@ f2 @@@ f)] == s*GCD[n, r]; Select[Range[10^6], q] (* Amiram Eldar, Nov 29 2021 *)
  • PARI
    A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
isA349745(n) = { my(s=sigma(n),u=A003961(n)); (n*gcd(s,u) == (s*gcd(n,u))); };

Formula

For all n >= 1, A007814(A000203(a(n))) = A007814(a(n)). [sigma preserves the 2-adic valuation of the terms of this sequence]

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.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Feb 16 2022

Keywords

Comments

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.

Examples

			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.

Links

Crossrefs

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).

Programs

  • PARI
    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)); };

A351555 a(n) is the number of odd prime factors of sigma(n) that do not divide A003961(n) and the valuation(n, p) is different from valuation(sigma(n), p).

Original entry on oeis.org

0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 2, 0, 0, 1, 0, 0, 1, 2, 1, 1, 1, 0, 2, 0, 1, 1, 2, 0, 1, 1, 2, 1, 1, 1, 1, 0, 1, 1, 0, 1, 2, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 0, 1, 2, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1, 1, 2, 0, 1, 2, 0
Offset: 1

Views

Author

Antti Karttunen, Feb 16 2022

Keywords

Links

Crossrefs

Cf. A000203, A003961, A351551, A351554 (positions of zeros), A351553 (even terms there), A351543.

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)); };

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

Views

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}.

A351552 Numbers k for which A351546(k) divides k, where A351546(n) is the largest unitary divisor of sigma(n) such that none of its prime factors divide A003961(n).

Original entry on oeis.org

1, 2, 10, 20, 24, 34, 40, 106, 120, 160, 216, 224, 260, 340, 408, 440, 480, 520, 580, 672, 680, 696, 792, 820, 1060, 1120, 1272, 1392, 1450, 1640, 1666, 1760, 1780, 1940, 2080, 2120, 2136, 2320, 2340, 2464, 2720, 2900, 3040, 3320, 3332, 3576, 3680, 3808, 3880, 3960, 4280, 4320, 4536, 4640, 4680, 5280, 5380, 5800, 5860
Offset: 1

Views

Author

Antti Karttunen, Feb 16 2022

Keywords

Comments

See comments in A351551.

Links

Crossrefs

Cf. A000203, A003961, A351546.
Not a subsequence of A351554.
Cf. A351551 (subsequence).

Programs

  • PARI
    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])); };
isA351552(n) = !(n%A351546(n));

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

Views

Author

Antti Karttunen, Feb 16 2022

Keywords

Comments

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

Links

Crossrefs

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.

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)); };
isA351553(n) = (!(n%2) && 0==A351555(n));

Extensions

Definition corrected by Antti Karttunen, Aug 27 2025
Showing 1-9 of 9 results.

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