A387711 -id:A387711 - cp's OEIS frontend

A387712 Primitive terms of A387711: numbers k for which A003959(k) > 2k, but for all whose proper divisors d|k, dA003959(d) <= 2d.

4, 18, 27, 30, 42, 45, 50, 63, 66, 70, 78, 102, 114, 138, 174, 186, 222, 246, 258, 282, 318, 354, 366, 375, 402, 426, 438, 474, 498, 525, 534, 582, 606, 618, 625, 642, 654, 678, 686, 735, 762, 786, 822, 825, 834, 894, 906, 942, 975, 978, 1002, 1038, 1074, 1078, 1086, 1089, 1146, 1158, 1182, 1194, 1210, 1266, 1274, 1275

Offset: 1

Antti Karttunen, Sep 06 2025

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A003959, A387711, A387713.
Positions of 1's in A387715.
Cf. also A091191, A337372.

A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A387715(n) = sumdiv(n,d,(A003959(d)>(2*d)));
is_A387712(n) = (1==A387715(n));

{k | A387715(k) == 1}.

A387713 Nonprimitive terms of A387711: numbers k for which A387715(k) > 1.

Original entry on oeis.org

8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 54, 56, 60, 64, 68, 72, 76, 80, 81, 84, 88, 90, 92, 96, 100, 104, 108, 112, 116, 120, 124, 126, 128, 132, 135, 136, 140, 144, 148, 150, 152, 156, 160, 162, 164, 168, 172, 176, 180, 184, 188, 189, 192, 196, 198, 200, 204, 208, 210, 212, 216, 220, 224, 225, 228, 232, 234

Offset: 1

Antti Karttunen, Sep 06 2025

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A003959.
Setwise difference A387711 \ A387712. Positions of terms > 1 in A387715.
Subsequence of A341610.

A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A387715(n) = sumdiv(n,d,(A003959(d)>(2*d)));
is_A387713(n) = (A387715(n)>1);

{k | A387715(k) > 1}.

A387710 Numbers k for which A003959(k) < 2*k, where A003959 is multiplicative with a(p^e) = (p+1)^e.

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 46, 47, 49, 51, 53, 55, 57, 58, 59, 61, 62, 65, 67, 69, 71, 73, 74, 75, 77, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 98, 99, 101, 103, 105, 106, 107, 109, 110, 111, 113, 115, 117, 118, 119, 121, 122, 123, 125

Offset: 1

Antti Karttunen, Sep 06 2025

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A003959, A387711.
Subsequence of A005100.
Subsequences: A000040, A001358\{4, 6}, A246281.
Positions of 0's in A387715.

A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
is_A387710(n) = (A003959(n)<(2*n));

A387715 Number of divisors d of n for which A003959(d) > 2*d, where A003959 is multiplicative with a(p^e) = (p+1)^e.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 3, 0, 1, 0, 2, 0, 0, 0, 4, 0, 0, 1, 2, 0, 1, 0, 4, 0, 0, 0, 4, 0, 0, 0, 4, 0, 1, 0, 2, 1, 0, 0, 6, 0, 1, 0, 2, 0, 3, 0, 4, 0, 0, 0, 5, 0, 0, 1, 5, 0, 1, 0, 2, 0, 1, 0, 7, 0, 0, 0, 2, 0, 1, 0, 6, 2, 0, 0, 5, 0, 0, 0, 4, 0, 4, 0, 2, 0, 0, 0, 8, 0, 0, 0, 4, 0, 1, 0, 4, 0

Offset: 1

Antti Karttunen, Sep 06 2025

Number of terms of A387711 that divide n.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A003959, A387711 (positions of terms > 0), A387712 (of 1's), A387713 (of terms > 1).

Cf. also A080224, A337345.

A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
is_A387711(n) = (A003959(n)>(2*n));
A387715(n) = sumdiv(n,d,is_A387711(d));

a(n) = Sum_{d|n} [A003959(d) > 2*d], where [ ] is the Iverson bracket.

A387721 Numbers k for which A107758(k) > 2*k, where A107758 is sigma+, multiplicative function with a(p^e) = 1+sigma(p^e).

Original entry on oeis.org

6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 30, 34, 36, 38, 40, 42, 44, 45, 46, 48, 50, 52, 54, 56, 58, 60, 62, 66, 68, 70, 72, 74, 75, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 105, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126, 130, 132, 134, 135, 136, 138, 140, 142, 144, 146, 148

Offset: 1

Antti Karttunen, Sep 06 2025

Note that in contrast to analogous sequences like A005101, A246282, A387711, here it is not guaranteed that multiples of any term are also terms. For example, 21 is a term, but 3*21 = 63 is in A052396. Also 15, 45, 75 are in this sequence, but their multiple 225 is in A387720.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A107758.
Complement of (A052396 U A387720).
Disjoint union of A387722 and A387723.
Positions of positive terms in A387725.
Cf. also A005101 (subsequence), A246282, A387711.

A107758(n) =  { my(f = factor(n)); prod(k=1, #f~, 1+sigma(f[k, 1]^f[k, 2])); };
is_A387721(n) = (A107758(n)>(2*n));

cp's OEIS Frontend

A387712 Primitive terms of A387711: numbers k for which A003959(k) > 2*k, but for all whose proper divisors d|k, dA003959(d) <= 2*d.

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A387713 Nonprimitive terms of A387711: numbers k for which A387715(k) > 1.

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A387710 Numbers k for which A003959(k) < 2*k, where A003959 is multiplicative with a(p^e) = (p+1)^e.

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A387715 Number of divisors d of n for which A003959(d) > 2*d, where A003959 is multiplicative with a(p^e) = (p+1)^e.

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A387721 Numbers k for which A107758(k) > 2*k, where A107758 is sigma+, multiplicative function with a(p^e) = 1+sigma(p^e).

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A387712 Primitive terms of A387711: numbers k for which A003959(k) > 2k, but for all whose proper divisors d|k, dA003959(d) <= 2d.