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 1-4 of 4 results.

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,", ")));

A332458 Odd numbers k, not powers of primes, such that sigma(k) == 2 modulo 8 and sigma(sigma(k)) == 6 modulo 8.

Original entry on oeis.org

4041, 5193, 15633, 25181, 25497, 26645, 30249, 36441, 36517, 40817, 47045, 53577, 54513, 85697, 87273, 92889, 93393, 94761, 95913, 97281, 111609, 129681, 136233, 143433, 151713, 161257, 162441, 163377, 165897, 171197, 177129, 210033, 212697, 213849, 222993, 228977, 229833, 232897, 238041, 245673, 246969, 250137, 260577
Offset: 1

Views

Author

Antti Karttunen, Feb 15 2020

Keywords

Links

Crossrefs

Cf. A000203, A000961, A051027.
Intersection of A228058 and A332457; intersection of A332456 and A332228.
Cf. also A332445.

Programs

  • Magma
    [k:k in [3..270000 by 2]| not IsPrimePower(k) and DivisorSigma(1,k) mod 8 eq 2 and DivisorSigma(1, DivisorSigma(1,k)) mod 8 eq 6]; // Marius A. Burtea, Feb 15 2020
  • PARI
    isA332458(n) = if(!(n%2)||isprimepower(n),0, my(s=sigma(n)); ((2==(s%8)) && (6==(sigma(s)%8))));

A332446 Numbers k for which A087808(sigma(k)) is equal to A087808(2*k).

Original entry on oeis.org

3, 6, 11, 19, 28, 43, 59, 67, 83, 107, 131, 139, 163, 179, 211, 216, 227, 251, 267, 283, 286, 307, 331, 347, 379, 419, 443, 467, 491, 496, 499, 523, 547, 563, 571, 587, 598, 619, 643, 659, 683, 691, 726, 739, 787, 811, 827, 859, 883, 907, 947, 971, 1019, 1051, 1091, 1123, 1163, 1171, 1187, 1259, 1283, 1291, 1307, 1427, 1451
Offset: 1

Views

Author

Antti Karttunen, Feb 14 2020

Keywords

Comments

Conjecture: includes all terms of A007520. - Bill McEachen, Dec 10 2023

Links

Crossrefs

Cf. A000203, A007520, A087808.
Subsequences: A000396, A332445.
Cf. A331751, A331752, A332208 for similar sequences.

Programs

A332465 Numbers n for which A269174(sigma(n)) is equal to 2*A269174(n).

Original entry on oeis.org

6, 28, 348, 496, 732, 886, 2924, 3573, 4972, 5448, 7544, 8128, 23388, 54842, 66928, 89200, 92296, 109786, 118064, 121552, 349512, 356488, 367472, 550432, 634784, 839984, 842452, 1234048, 1561408, 1797496, 2154584, 2364832, 2788808, 2927992, 3451456, 3585328, 5952364, 5991852, 6687136, 8238752, 10594336, 11210712, 11261020
Offset: 1

Views

Author

Antti Karttunen, Feb 16 2020

Keywords

Comments

Numbers n such that A332464(n) is equal to A269174(2*n).
There are only three odd terms <= 2^32 among the first 113 terms of this sequence: 3573, 29255157, 936109557. Because A269174 preserves the 2-adic valuation of its argument, all such odd terms are of the form 4m+1, and must be present in A191218. Incidentally, these three terms are also present in A228058, but not in A332227.
See from the graph how unevenly the terms appear. Compare also the scatter plots of A269174 and A332464, also of a similar sequence A332445.

Examples

			          k   factorization        sigma(k)     A269174(sigma(k)) = A269174(2*k)
        348 = 2^2 * 3 * 29         840          2008,
       3573 = 3^2 * 397            5174         15486,
   29255157 = 3^2 * 3250573        42257462     126737534,
  936109557 = 3^2 * 104012173      1352158262   4055424126.

Links

Crossrefs

Cf. A000203, A269174, A332464.
Cf. A000396 (a subsequence).
Cf. also A191218, A228058, A332227, A332445, A332446.

Programs

  • Mathematica
    b[n_] := BitAnd[BitOr[n, 2n], BitOr[BitXor[n, 2n], BitXor[n, 4n]]];
okQ[n_] := b[DivisorSigma[1, n]] == 2 b[n];
Reap[For[n = 1, n <= 12*10^6, n++, If[okQ[n], Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Feb 23 2020 *)
  • PARI
    A269174(n) = bitand(bitor(n,n<<1),bitor(bitxor(n,n<<1),bitxor(n,n<<2)));
isA332465(n) = (A269174(sigma(n))==2*A269174(n));
Showing 1-4 of 4 results.

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