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 1-10 of 13 results. Next

A351446 Numbers k for which A003958(sigma(k)) = A003958(k), where A003958 is multiplicative with a(p^e) = (p-1)^e and sigma is the sum of divisors function.

Original entry on oeis.org

1, 6, 10, 26, 28, 49, 54, 74, 122, 126, 146, 294, 314, 386, 408, 490, 496, 554, 626, 680, 794, 842, 914, 1082, 1226, 1232, 1274, 1322, 1346, 1466, 1514, 1560, 1754, 1768, 1994, 2186, 2306, 2402, 2426, 2474, 2642, 2646, 2762, 2906, 3242, 3314, 3360, 3506, 3626, 3672, 3746, 3808, 3866, 3986, 4034
Offset: 1

Views

Author

Antti Karttunen, Feb 12 2022

Keywords

Comments

Numbers k for which A351442(k) = A003958(k), or equally, for which k = A351444(k) = A322582(k) + A351442(k).

Links

Crossrefs

Cf. A000203, A003958, A322582, A351442, A351447.
Fixed points of A351444, positions of zeros in A351445.
Subsequences: A000396, A351443 (odd terms), A351440, A336702 (numbers k for which A064989(sigma(k)) = A064989(k)).

Programs

A351445 a(n) = A003958(sigma(n)) - A003958(n), where A003958 is multiplicative with a(p^e) = (p-1)^e and sigma is the sum of divisors function.

Original entry on oeis.org

0, 1, -1, 5, -2, 0, -5, 7, 8, 0, -8, 4, -6, -4, -6, 29, -12, 20, -14, 8, -11, -6, -20, 6, 14, 0, -4, 0, -20, -4, -29, 23, -18, -8, -22, 68, -18, -10, -18, 12, -28, -10, -32, 2, 8, -18, -44, 28, 0, 44, -28, 24, -44, 0, -36, 2, -32, -12, -50, 4, -30, -28, -12, 125, -36, -16, -50, 8, -42, -20, -66, 92, -36, 0
Offset: 1

Views

Author

Antti Karttunen, Feb 12 2022

Keywords

Links

Crossrefs

Cf. A000203, A003958, A351442, A351444, A351457 [= a(A003961(n))].
Cf. A351446 (positions of zeros), A351443 (odd terms there).
Cf. also A348736.

Programs

Formula

a(n) = A351442(n) - A003958(n) = A351444(n) - n.

A351443 Odd numbers k for which A003958(sigma(k)) = A003958(k), where A003958 is multiplicative with a(p^e) = (p-1)^e and sigma is the sum of divisors function.

Original entry on oeis.org

1, 49, 40905, 106353, 140211, 275301, 302697, 499041, 597213, 1094913, 1284417, 1578933, 2004345, 2266137, 2560653, 3247857, 3444201, 3738717, 4425921, 5014953, 5123817, 5211297, 5407641, 5505813, 5996673, 6193017, 6870339, 7174737, 8156457, 8941833, 9432693, 9825381, 9923553
Offset: 1

Views

Author

Antti Karttunen, Feb 12 2022

Keywords

Comments

Odd numbers k for which A351442(k) = A003958(k), or equally, for which k = A351444(k) = A322582(k) + A351442(k).
The 13th term, 2004345, is one of the rare abundant numbers (A005101, A005231) in this sequence.

Links

Crossrefs

Odd terms in A351446.
Cf. A000203, A003958, A005101, A005231, A322582, A351442, A351444, A351445.
These terms doubled form a subsequence of A351447.

Programs

A351444 a(n) = n - A003958(n) + A003958(sigma(n)), where A003958 is multiplicative with a(p^e) = (p-1)^e and sigma is the sum of divisors function.

Original entry on oeis.org

1, 3, 2, 9, 3, 6, 2, 15, 17, 10, 3, 16, 7, 10, 9, 45, 5, 38, 5, 28, 10, 16, 3, 30, 39, 26, 23, 28, 9, 26, 2, 55, 15, 26, 13, 104, 19, 28, 21, 52, 13, 32, 11, 46, 53, 28, 3, 76, 49, 94, 23, 76, 9, 54, 19, 58, 25, 46, 9, 64, 31, 34, 51, 189, 29, 50, 17, 76, 27, 50, 5, 164, 37, 74, 73, 82, 19, 66, 5, 136
Offset: 1

Views

Author

Antti Karttunen, Feb 12 2022

Keywords

Links

Crossrefs

Cf. A000203, A003958, A322582, A351442, A351445.
Cf. A351446 (fixed points), A351443 (odd terms there).

Programs

Formula

a(n) = A322582(n) + A351442(n) = n - A003958(n) + A003958(sigma(n)).
a(n) = n + A351445(n).

A351456 a(n) = A003958(sigma(A003961(n))), where A003958 is multiplicative with a(p^e) = (p-1)^e, A003961 multiplicative with a(prime(k)^e) = prime(1+k)^e, and sigma is the sum of divisors function.

Original entry on oeis.org

1, 1, 2, 12, 1, 2, 2, 4, 30, 1, 6, 24, 4, 2, 2, 100, 4, 30, 2, 12, 4, 6, 8, 8, 36, 4, 24, 24, 1, 2, 18, 72, 12, 4, 2, 360, 12, 2, 8, 4, 10, 4, 2, 72, 30, 8, 8, 200, 108, 36, 8, 48, 8, 24, 6, 8, 4, 1, 30, 24, 16, 18, 60, 1092, 4, 12, 4, 48, 16, 2, 36, 120, 4, 12, 72, 24, 12, 8, 12, 100, 700, 10, 16, 48, 4, 2, 2, 24
Offset: 1

Views

Author

Antti Karttunen, Feb 12 2022

Keywords

Links

Crossrefs

Cf. A000203, A003958, A003961, A003973, A151800, A339905, A351442, A351457.
Cf. also A326042.

Programs

  • PARI
    A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A351456(n) = A003958(sigma(A003961(n)));

Formula

Multiplicative with a(p^e) = A003958(1 + q + ... + q^e), where q = nextPrime(p) = A151800(p).
a(n) = A003958(A003973(n)) = A351442(A003961(n)) = A003958(A000203(A003961(n))).
a(n) = A351457(n) + A339905(n).

A353792 a(n) = A003958(sigma(n)) * A064989(sigma(n)).

Original entry on oeis.org

1, 4, 1, 30, 4, 4, 1, 48, 132, 16, 4, 30, 30, 4, 4, 870, 16, 528, 12, 120, 1, 16, 4, 48, 870, 120, 12, 30, 48, 16, 1, 480, 4, 64, 4, 3960, 306, 48, 30, 192, 120, 4, 70, 120, 528, 16, 4, 870, 1224, 3480, 16, 900, 64, 48, 16, 48, 12, 192, 48, 120, 870, 4, 132, 14238, 120, 16, 208, 480, 4, 16, 16, 6336, 1116, 1224, 870
Offset: 1

Views

Author

Antti Karttunen, May 11 2022

Keywords

Links

Crossrefs

Cf. A003958, A064989, A350073, A351442, A353791, A353793, A353794 [= a(A003961(n))].
Cf. A046528 (positions of 1's).
Cf. also A353750.

Programs

  • PARI
    A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
A064989(n) = { my(f=factor(n>>valuation(n,2))); for(i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
A353792(n) = { my(s=sigma(n)); (A003958(s)*A064989(s)); };
  • PARI
    A353792(n) = { my(f=factor(n),s); prod(i=1, #f~, s = sigma(f[i,1]^f[i,2]); A003958(s)*A064989(s)); };

Formula

Multiplicative with a(p^e) = A003958(1 + p + ... + p^e) * A064989(1 + p + ... + p^e).
a(n) = A353791(A000203(n)).
a(n) = A351442(n) * A350073(n) = A003958(A000203(n)) * A064989(A000203(n)).

A351440 Numbers k for which A003958(sigma(k)) + A064989(sigma(k)) is equal to A003958(k) + A064989(k).

Original entry on oeis.org

1, 6, 28, 496, 8128, 30240, 32760, 240408, 2178540, 6828720, 13042080, 23569920, 33550336, 42402048, 45532800, 142990848, 1379454720
Offset: 1

Views

Author

Antti Karttunen, Feb 12 2022

Keywords

Links

Crossrefs

Cf. A000203, A003958, A064989, A351442.
Subsequence of A351446.
Subsequences: A000396, A336702.

Programs

  • PARI
    A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
A064989(n) = { my(f = factor(n>>valuation(n,2))); for(i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
isA351440(n) = { my(s=sigma(n)); ((A003958(s)+A064989(s)) == (A003958(n)+A064989(n))); };

A351447 Numbers k for which A003958(sigma(k)) = 2*A003958(k), where A003958 is multiplicative with a(p^e) = (p-1)^e and sigma is the sum of divisors function.

Original entry on oeis.org

2, 98, 120, 136, 312, 520, 672, 888, 1080, 1120, 1464, 1480, 1752, 2440, 2520, 2808, 2912, 2920, 3420, 3768, 3848, 4632, 5880, 6048, 6280, 6344, 6552, 6648, 6664, 7512, 7592, 7720, 7992, 8181, 8288, 8892, 9528, 10104, 10968, 11080, 12464, 12520, 12984, 13176, 13664, 14712, 15288
Offset: 1

Views

Author

Antti Karttunen, Feb 12 2022

Keywords

Comments

Numbers k such that A351442(k) = 2*A003958(k).
In contrast, numbers x for which A064989(sigma(x)) = 2*A064989(x) seem to consist just of {2} followed by A005820: 2, 120, 672, 523776, ..., etc, which (also) contains as its subsequence all the odd terms of A336702 multiplied by 2.

Links

Crossrefs

Cf. A000203, A003958, A064989, A351442.
Subsequences: A005820 (3-perfect numbers), odd terms of A336702 doubled, the terms of A351443 doubled (2, 98, 81810, ...), A351448 (odd terms in this sequence).

Programs

A387157 a(n) = A173557(sigma(n)), 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, 2, 1, 6, 2, 2, 1, 8, 12, 2, 2, 6, 6, 2, 2, 30, 2, 24, 4, 12, 1, 2, 2, 8, 30, 12, 4, 6, 8, 2, 1, 12, 2, 2, 2, 72, 18, 8, 6, 8, 12, 2, 10, 12, 24, 2, 2, 30, 36, 60, 2, 6, 2, 8, 2, 8, 4, 8, 8, 12, 30, 2, 12, 126, 12, 2, 16, 12, 2, 2, 2, 96, 36, 36, 30, 24, 2, 12, 4, 60, 10, 12, 12, 6, 2, 20, 8, 8, 8, 24, 6, 12, 1, 2, 8
Offset: 1

Views

Author

Antti Karttunen, Aug 19 2025

Keywords

Links

Crossrefs

Cf. A000203, A003958, A080398, A173557, A387158 (positions where equal to A173557(n)).
Cf. also A351442.

Programs

  • Mathematica
    A387157[n_] := If[n == 1, 1, Times @@ (FactorInteger[DivisorSigma[1, n]][[All, 1]] - 1)];
Array[A387157, 100] (* Paolo Xausa, Aug 20 2025 *)
  • PARI
    A387157(n) = factorback(apply(p -> p-1,factor(sigma(n))[,1]));

Formula

a(n) = A003958(A080398(n)).

A351457 a(n) = A351456(n) - A339905(n).

Original entry on oeis.org

0, -1, -2, 8, -5, -6, -8, -4, 14, -11, -6, 8, -12, -18, -22, 84, -14, -2, -20, -12, -36, -18, -20, -24, 0, -28, -40, -16, -29, -46, -18, 40, -36, -32, -58, 296, -28, -42, -56, -44, -32, -76, -44, 24, -66, -48, -44, 136, 8, -36, -64, -16, -50, -104, -66, -72, -84, -59, -30, -72, -50, -54, -100, 1028, -92, -84, -66
Offset: 1

Views

Author

Antti Karttunen, Feb 12 2022

Keywords

Links

Crossrefs

Cf. A000203, A003958, A003961, A003973, A339905, A351442, A351456.
Cf. also A348736.

Programs

  • PARI
    A339905(n) = if(1==n,n,my(f=factor(n)); for(i=1,#f~,f[i,1] = nextprime(1+f[i,1])-1); factorback(f));
A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A351456(n) = A003958(sigma(A003961(n)));
A351457(n) = (A351456(n) - A339905(n));

Formula

a(n) = A351445(A003961(n)) = A351456(n) - A339905(n).
Showing 1-10 of 13 results. Next

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