A351445 -id:A351445 - 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).

A353636 Difference between phi(sigma(n)) and phi(n).

Original entry on oeis.org

0, 1, 0, 4, -2, 2, -2, 4, 6, 2, -6, 8, -6, 2, 0, 22, -10, 18, -10, 4, 4, 2, -14, 8, 10, 0, -2, 12, -20, 16, -14, 20, -4, 2, -8, 60, -18, -2, 0, 8, -28, 20, -22, 4, 0, 2, -30, 44, -6, 40, -8, 18, -34, 14, -16, 8, -4, -4, -42, 32, -30, 2, 12, 94, -24, 28, -34, 4, -12, 24, -46, 72, -36, 0, 20, 12, -28, 24, -46, 28, 56

Offset: 1

Antti Karttunen, May 04 2022

Antti Karttunen, Table of n, a(n) for n = 1..10000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences related to sigma(n)

Cf. A000010, A000203, A007431, A062401, A353643, A353647.
Cf. A006872 (positions of zeros), A353637 (their characteristic function).
Cf. A353682 (positions of terms >= 0), A353683 (of terms > 0), A353685 (of terms <= 0), A353686 (of negative terms).
Cf. also A351445.

a[n_] := EulerPhi[DivisorSigma[1, n]] - EulerPhi[n]; Array[a, 100] (* Amiram Eldar, May 06 2022 *)

A353636(n) = (eulerphi(sigma(n))-eulerphi(n));

a(n) = A062401(n) - A000010(n) = A000010(A000203(n)) - A000010(n).

a(n) = Sum_{d|n} (A353647(d) - A007431(d)).

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)));

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

Antti Karttunen, Feb 12 2022

nonn

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

Cf. A000203, A003958, A322582, A351442, A351445.

Cf. A351446 (fixed points), A351443 (odd terms there).

A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
A351444(n) = (n - A003958(n) + A003958(sigma(n)));

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

a(n) = n + A351445(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

Antti Karttunen, Feb 12 2022

sign

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

Cf. also A348736.

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));

a(n) = A351445(A003961(n)) = A351456(n) - A339905(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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A353636 Difference between phi(sigma(n)) and phi(n).

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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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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.

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

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