A351447 -id:A351447 - cp's OEIS frontend

A351442 a(n) = A003958(sigma(n)), where A003958 is multiplicative with a(p^e) = (p-1)^e and sigma is the sum of divisors function.

1, 2, 1, 6, 2, 2, 1, 8, 12, 4, 2, 6, 6, 2, 2, 30, 4, 24, 4, 12, 1, 4, 2, 8, 30, 12, 4, 6, 8, 4, 1, 24, 2, 8, 2, 72, 18, 8, 6, 16, 12, 2, 10, 12, 24, 4, 2, 30, 36, 60, 4, 36, 8, 8, 4, 8, 4, 16, 8, 12, 30, 2, 12, 126, 12, 4, 16, 24, 2, 4, 4, 96, 36, 36, 30, 24, 2, 12, 4, 60, 100, 24, 12, 6, 8, 20, 8

Offset: 1

Antti Karttunen, Feb 12 2022

Question: Are there more fixed points than 1, 2, 8, 128, 288, 720, 32768, 29719872, ..., 2147483648 ?

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to sigma(n)

Cf. A000203, A003958, A322582, A339905, A351443, A351444, A351445, A351446, A351447, A351448, A351456.

Cf. also A348512.

A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
A351442(n) = A003958(sigma(n));

Multiplicative with a(p^e) = A003958(1 + p + ... + p^e).
a(n) = A003958(A000203(n)).
a(n) = A351444(n) - A322582(n) = A351445(n) + A003958(n).

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

Antti Karttunen, Feb 12 2022

nonn

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

Index entries for sequences related to sigma(n)

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)).

A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
isA351446(n) = (A003958(sigma(n))==A003958(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

Antti Karttunen, Feb 12 2022

nonn

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.

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

A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
A351442(n) = A003958(sigma(n));
isA351443(n) = ((n%2) && (A351442(n)==A003958(n)));

A351448 Odd 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

8181, 400869, 1507005, 3918213, 11151837, 22002273, 26669007, 47319957, 58170393, 73843245, 75825981, 83488077, 94338513, 108277641, 119656197, 126889821, 137740257, 163057941, 184758813, 191992437, 199226061, 202842873, 204768225, 220926933, 228160557, 258457473, 264328677, 277602471, 300496797

Offset: 1

Antti Karttunen, Feb 12 2022

nonn

Odd numbers k such that A351442(k) = 2*A003958(k).
Any hypothetical odd term of A005820, if such a term exists, should appear in this sequence, in A347391, and in A016754 (odd squares).
None of the first 33 terms is a square, and all of them except 75825981 and 204768225 are multiples of 81. Note that 81 is one of the terms of A008848 (and of A231484), squares whose sum of divisors is also square (with A000203(81) = 121).

Index entries for sequences related to sigma(n)

Odd terms in A351447.

Cf. A000203, A003958, A005820, A008848, A016754, A231484, A339905, A347391, A351442.

A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
isA351448(n) = (n%2 && (A003958(sigma(n)) == 2*A003958(n)));

cp's OEIS Frontend

A351442 a(n) = A003958(sigma(n)), where A003958 is multiplicative with a(p^e) = (p-1)^e and sigma is the sum of divisors function.

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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.

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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.

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A351448 Odd 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.

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