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.

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A332225 Numbers k > 1 for which A048675(A332223(k)) is equal to 2*A048675(k).

Original entry on oeis.org

4, 9, 12, 20, 44, 52, 60, 108, 124, 125, 132, 140, 156, 172, 188, 204, 236, 300, 308, 396, 412, 436, 476, 492, 612, 644, 700, 836, 876, 884, 891, 924, 972, 980, 1004, 1044, 1092, 1100, 1116, 1148, 1188, 1196, 1236, 1260, 1268, 1292, 1300, 1308, 1372, 1380, 1476, 1620, 1628, 1724, 1860, 1900, 2140, 2244, 2324, 2356, 2444, 2460, 2652, 2660, 2700
Offset: 1

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Author

Antti Karttunen, Feb 12 2020

Keywords

Comments

Numbers k > 1 such that A332224(A156552(k)) = A087808(sigma(A156552(k))) is equal to 2*A048675(k) = A048675(k^2).
Notably, of the first 150 terms (4 .. 9996), 156 = 2^2 * 3 * 13 is the only even term that does not map to a prime, as A156552(156) = 267 = 3*89 (and sigma(267) = 360 = 4*90).
Although sigma(A156552(k)) = A323243(k) is a multiple of 4 for most of the terms k present in this sequence, there are exceptions, for example 840350 = A005940(1+A332445(1)) = 2^1 * 5^2 * 7^5 is one, as A048675(A332223(840350)) = 98 = 2*A048675(840350) and A323243(840350) = 2394 == 2 (mod 4).

Links

Crossrefs

Cf. A000203, A048675, A156552, A323243, A332223, A332224, A332229, A332445, A332446.
Cf. A324201 (a subsequence).

Programs

  • PARI
    for(n=2,2048,if(A048675(A332223(n))==2*A048675(n),print1(n,", ")))
  • PARI
    \\ To find all terms < 10000:
v156552sigs = readvec("a156552.txt"); \\ Use the factorization file for A156552 prepared by Hans Havermann, available at https://oeis.org/A156552/a156552.txt
A323243(n) = if(n<=2,n-1,my(prsig=v156552sigs[n],ps=prsig[1],es=prsig[2]); prod(i=1,#ps,((ps[i]^(1+es[i]))-1)/(ps[i]-1)));
A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
A087808(n) = if(n<1, 0, if(n%2==0, 2*A087808(n/2), A087808((n-1)/2)+1));
isA322225(n) = (A087808(A323243(n)) == 2*A048675(n));
for(n=2,10000,if(isA322225(n),print1(n,", ")));

A332228 Odd numbers n, not powers of primes, such that sigma(n) is congruent to 2 modulo 8.

Original entry on oeis.org

153, 325, 369, 657, 725, 801, 833, 845, 873, 925, 1017, 1233, 1325, 1377, 1445, 1525, 1737, 2009, 2057, 2097, 2169, 2313, 2525, 2529, 2725, 2817, 2925, 3033, 3177, 3321, 3577, 3609, 3681, 3725, 3757, 3897, 3925, 4041, 4113, 4205, 4325, 4361, 4525, 4689, 4753, 4901, 4925, 4961, 5121, 5193, 5337, 5409, 5537, 5553, 5725
Offset: 1

Views

Author

Antti Karttunen, Feb 13 2020

Keywords

Comments

Proof that any odd perfect number, if such numbers exist at all, has to reside in this sequence: As all terms in A228058 are = 1 modulo 4 (their binary expansion ends as "01"), and taking sigma of an odd perfect number would multiply it by two (shift one bit-position left), the base-2 expansion of that result would end as "010", i.e., sigma(k) modulo 8 should be 2 (not 6) for such numbers k.

Links

Crossrefs

Cf. A000203, A332229.
Subsequence of A228058, of A332226 and of A332227.

Programs

  • PARI
    isA332228(n) = ((n%2)&&!isprimepower(n)&&2==(sigma(n)%8));
Showing 1-2 of 2 results.

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