Max Alekseyev - cp's OEIS frontend

A387352 Numbers m with deficiency 32: sigma(m) - 2*m = -32.

250, 376, 1276, 12616, 20536, 396916, 801376, 1297312, 8452096, 33721216, 40575616, 59376256, 89397016, 99523456, 101556016, 150441856, 173706136, 269096704, 283417216, 500101936, 1082640256, 1846506832, 15531546112, 34675557856, 136310177392, 136783784608

Offset: 1

Max Alekseyev, Aug 27 2025

Contains numbers 2^(k-1)*(2^k + 31) for k in A247952.

Max Alekseyev, Table of n, a(n) for n = 1..54

Deficiency k: A191363 (k=2), A125246 (k=4), A141548 (k=6), A125247 (k=8), A101223 (k=10), A141549 (k=12), A141550 (k=14), A125248 (k=16), A223608 (k=18), A223607 (k=20), A223606 (k=22), A385255(k=24), A275702 (k=26), A275997 (k=64).
Abundance k: A088831 (k=2), A088832 (k=4), A087167 (k=6), A088833 (k=8), A223609 (k=10), A141545 (k=12), A141546 (k=14), A141547 (k=16), A223610 (k=18), A223611 (k=20), A223612 (k=22), A223613 (k=24), A275701 (k=26), A175989 (k=32), A275996 (k=64), A292626 (k=128).
Cf. A247952.

A385255 Numbers m whose deficiency is 24: sigma(m) - 2*m = -24.

Original entry on oeis.org

124, 9664, 151115727458150838697984

Offset: 1

Max Alekseyev, Jul 29 2025

Contains numbers 2^(k-1)*(2^k + 23) for k in A057203. First three terms have this form.

Deficiency k: A191363 (k=2), A125246 (k=4), A141548 (k=6), A125247 (k=8), A101223 (k=10), A141549 (k=12), A141550 (k=14), A125248 (k=16), A223608 (k=18), A223607 (k=20), A223606 (k=22), A275702 (k=26).
Abundance k: A088831 (k=2), A088832 (k=4), A087167 (k=6), A088833 (k=8), A223609 (k=10), A141545 (k=12), A141546 (k=14), A141547 (k=16), A223610 (k=18), A223611 (k=20), A223612 (k=22), A223613 (k=24), A275701 (k=26).
Cf. A057203.

A386399 Number of forests with at most n unlabeled nodes.

Original entry on oeis.org

1, 2, 4, 7, 13, 23, 43, 80, 156, 309, 638, 1348, 2949, 6607, 15206, 35720, 85625, 208588, 515787, 1291316, 3269194, 8355832, 21539988, 55942920, 146271594, 384746580, 1017522228, 2704227858, 7219183490, 19351410860, 52068524665, 140588391713, 380824067016

Offset: 0

Max Alekseyev, Jul 20 2025

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Partial sums of A005195.

Cf. A000272, A002807, A006897, A054548, A137917, A367863, A372176.

G.f.: exp(sum_{k>0} B(x^k)/k ) / (1-x), where B(x) = x + x^2 + x^3 + 2*x^4 + 3*x^5 + 6*x^6 + 11*x^7 + ... = C(x)-1 and C is the g.f. for A000055.

A384035 Number of vector differences between two permutations of order n, up to multiplication by positive rational numbers and permutations of the components.

Original entry on oeis.org

1, 1, 2, 4, 13, 49, 228, 1034, 5133, 25710, 133872, 708976, 3846150, 21170077, 118429072, 670537495

Offset: 0

Max Alekseyev, May 17 2025

			For n = 3, there are A019589(3) = 5 difference vectors up to permutation of components: (-2, 0, 2), (-2, 1, 1), (-1, -1, 2), (-1, 0, 1), and (0, 0, 0). However, (-2, 0, 2) and (-1, 0, 1) are the same up to a factor 2. Hence, a(3) = 4.

Cf. A019589, A175176, A362968, A381243, A381244, A381339.

A383926 Powers m^p with prime p, producing primes (m^(p^2) - 1)/(m^p - 1) in A383925.

Original entry on oeis.org

4, 16, 36, 8, 100, 196, 256, 400, 576, 676, 27, 1296, 1600, 2916, 3136, 4356, 5476, 7056, 8100, 8836, 12100, 13456, 14400, 15376, 15876, 16900, 17956, 21316, 22500, 24336, 25600, 28900, 30976, 32400, 33856, 41616, 42436, 44100, 50176, 52900, 55696, 57600, 62500, 65536, 67600, 69696

Offset: 1

Max Alekseyev, May 15 2025

nonn

Permutation of A383923.

Cf. A383924, A383925.

a(n) = m^p with a prime p such that (a(n)^p - 1)/(a(n) - 1) = A383925(n).

A383925 Primes of the form (m^(p^2) - 1)/(m^p - 1) with prime p and integer m >= 2.

Original entry on oeis.org

5, 17, 37, 73, 101, 197, 257, 401, 577, 677, 757, 1297, 1601, 2917, 3137, 4357, 5477, 7057, 8101, 8837, 12101, 13457, 14401, 15377, 15877, 16901, 17957, 21317, 22501, 24337, 25601, 28901, 30977, 32401, 33857, 41617, 42437, 44101, 50177, 52901, 55697, 57601, 62501, 65537, 67601, 69697

Offset: 1

Max Alekseyev, May 15 2025

nonn

Corresponding values of m^p are listed in A383926.
Permutation of A383924.
Cf. A383923.

A383924 Primes of the form (m^(p^2) - 1)/(m^p - 1) with a prime p, sorted with respect to the value of m^p.

Original entry on oeis.org

5, 73, 17, 757, 37, 101, 197, 257, 401, 513, 577, 677, 1297, 1772893, 1601, 2917, 3137, 4357, 5477, 7057, 64008001, 8101, 8837, 85775383, 12101, 13457, 14401, 15377, 15877, 16901, 308933353, 17957, 21317, 22501, 24337, 25601, 729027001, 28901, 30977, 32401, 33857, 41617, 42437, 44101

Offset: 1

Max Alekseyev, May 15 2025

nonn

The corresponding values of m^p are listed in A383923.
Permutation of A383925.
Cf. A383926.

A383923 Numbers of the form m^p where both p and (m^(p^2) - 1)/(m^p - 1) are prime.

Original entry on oeis.org

4, 8, 16, 27, 36, 100, 196, 256, 400, 512, 576, 676, 1296, 1331, 1600, 2916, 3136, 4356, 5476, 7056, 8000, 8100, 8836, 9261, 12100, 13456, 14400, 15376, 15876, 16900, 17576, 17956, 21316, 22500, 24336, 25600, 27000, 28900, 30976, 32400, 33856, 41616, 42436, 44100, 50176, 52900

Offset: 1

Max Alekseyev, May 15 2025

nonn

Primes (m^(p^2) - 1)/(m^p - 1) are listed in A383924.
Permutation of A383926.
Cf. A383925.

A381339 Number of vector differences between two permutations of order n, up to multiplication by nonzero rational numbers and permutations of the components.

Original entry on oeis.org

1, 1, 2, 3, 9, 28, 128, 539, 2651, 13000, 67466, 355381, 1926343, 10590537, 59234734, 335302599

Offset: 0

Max Alekseyev, Feb 20 2025

Nonzero difference vectors are associated with their images in the projective space, and in addition we do not distinguish vectors that can be permuted one into the other. In the affine space, their number (including zero vector) is given by A019589, implying that a(n) <= A019589(n). Nonzero difference vectors in the projective space are counted by A381243.

			For n = 3, there are A019589(3) = 5 difference vectors up to permutation of components: (-2, 0, 2), (-2, 1, 1), (-1, -1, 2), (-1, 0, 1), and (0, 0, 0). However, (-2, 0, 2) and (-1, 0, 1) are the same up to a factor 2, and (-2, 1, 1) and (-1, -1, 2) are the same up to negation and reversing the order. Hence, a(3) = 3.

Cf. A019589, A175176, A362968, A381243, A381244.

A381243 Number of hyperplanes defined by the nonzero differences of two permutations of order n.

Original entry on oeis.org

0, 0, 1, 6, 85, 1370, 30481, 778610, 24409645, 881325366, 36635553601, 1713454403210, 89415912126223, 5143372266050837, 323667807885619744, 22112062644980805684

Offset: 0

Max Alekseyev, Feb 17 2025

Each of A175176(n) - 1 nonzero differences between two permutations (viewed as vectors) defines a hyperplane in the n-dimensional space. a(n) gives the number of pairwise distinct hyperplanes among them.

Cf. A019589, A175176, A362968, A381244, A381339.

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User: Max Alekseyev

A387352 Numbers m with deficiency 32: sigma(m) - 2*m = -32.

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A385255 Numbers m whose deficiency is 24: sigma(m) - 2*m = -24.

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A386399 Number of forests with at most n unlabeled nodes.

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A384035 Number of vector differences between two permutations of order n, up to multiplication by positive rational numbers and permutations of the components.

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A383926 Powers m^p with prime p, producing primes (m^(p^2) - 1)/(m^p - 1) in A383925.

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A383925 Primes of the form (m^(p^2) - 1)/(m^p - 1) with prime p and integer m >= 2.

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A383924 Primes of the form (m^(p^2) - 1)/(m^p - 1) with a prime p, sorted with respect to the value of m^p.

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A383923 Numbers of the form m^p where both p and (m^(p^2) - 1)/(m^p - 1) are prime.

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A381339 Number of vector differences between two permutations of order n, up to multiplication by nonzero rational numbers and permutations of the components.

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A381243 Number of hyperplanes defined by the nonzero differences of two permutations of order n.

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