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.

Previous Showing 41-44 of 44 results.

A252892 Primes p such that sigma(p) = 1 + p is a partition number (sorted increasingly).

Original entry on oeis.org

2, 29, 41, 6841, 37337, 53173, 105557, 124753, 614153, 26543659, 541946239, 2841940499, 3519222691, 30388671977, 6622987708039, 3925922161489421, 1089657644424399781, 9147679068859117601, 13196258966925435701, 505499305314204629557, 2715220650772245313219
Offset: 1

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Author

Omar E. Pol, Dec 24 2014

Keywords

Comments

Primes of the form p(k) - 1, where p(k) is a partition number (see A000040).
Primes in A000065. Intersection of A000040 and A000065.
Primes in A252891. Intersection of A000040 and A252891.

Examples

			41 is in the sequence because 41 is prime and the sum of divisors of 41 is 1 + 41 = 42 and 42 is the partition number of 10.

Links

Crossrefs

Cf. A000040, A000041, A000065, A000203, A052001, A252891.

Programs

  • PARI
    lista() = {v = readvec("b000041.txt"); for (n=1, #v, if (isprime(p=v[n]-1), print1(p, ", ")););} \\ Michel Marcus, Dec 29 2014

Extensions

More terms from Michel Marcus, Dec 28 2014
Edited by Wolfdieter Lang, Jan 14 2015

A304627 a(n) = [x^n] Product_{k>=1} (1 + x^k)*(1 - x^(n*k))/((1 - x^k)*(1 + x^(n*k))).

Original entry on oeis.org

1, 0, 2, 6, 12, 22, 38, 62, 98, 152, 230, 342, 502, 726, 1038, 1470, 2060, 2862, 3946, 5398, 7334, 9902, 13286, 17726, 23526, 31064, 40822, 53406, 69566, 90246, 116622, 150142, 192610, 246254, 313806, 398638, 504884, 637590, 802934, 1008446, 1263270, 1578526, 1967694, 2447062
Offset: 0

Views

Author

Ilya Gutkovskiy, May 15 2018

Keywords

Links

Crossrefs

Cf. A000065, A015128, A052839, A080054, A098151, A103258, A111133, A138526, A262968.

Programs

  • Mathematica
    Table[SeriesCoefficient[Product[(1 + x^k) (1 - x^(n k))/((1 - x^k) (1 + x^(n k))) , {k, 1, n}], {x, 0, n}], {n, 0, 43}]
Table[SeriesCoefficient[Product[(1 + x^k)/(1 - x^k), {k, 1, n - 1}], {x, 0, n}], {n, 0, 43}]
Join[{1}, Table[SeriesCoefficient[EllipticTheta[4, 0, x^n]/EllipticTheta[4, 0, x], {x, 0, n}], {n, 43}]]
nmax = 43; CoefficientList[Series[1/EllipticTheta[4, 0, x] - 2 x/(1 - x), {x, 0, nmax}], x]

Formula

G.f.: 1/theta_4(x) - 2*x/(1 - x), where theta_4() is the Jacobi theta function.
a(n) ~ exp(Pi*sqrt(n)) / (8*n). - Vaclav Kotesovec, May 19 2018

A357476 Number of partitions of n into two or more powers of 2.

Original entry on oeis.org

0, 0, 1, 2, 3, 4, 6, 6, 9, 10, 14, 14, 20, 20, 26, 26, 35, 36, 46, 46, 60, 60, 74, 74, 94, 94, 114, 114, 140, 140, 166, 166, 201, 202, 238, 238, 284, 284, 330, 330, 390, 390, 450, 450, 524, 524, 598, 598, 692, 692, 786, 786, 900, 900, 1014, 1014, 1154, 1154, 1294, 1294, 1460
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 02 2022

Keywords

Crossrefs

Cf. A000065, A018819, A209229, A357534.

Formula

a(n) = A018819(n) - A209229(n) for n > 0.

A364058 Heinz numbers of integer partitions with median > 1. Numbers whose multiset of prime factors has median > 2.

Original entry on oeis.org

3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 50, 51, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 81, 82, 83, 84, 85, 86
Offset: 1

Views

Author

Gus Wiseman, Jul 14 2023

Keywords

Comments

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

Examples

			The terms together with their prime indices begin:
     3: {2}        23: {9}          42: {1,2,4}
     5: {3}        25: {3,3}        43: {14}
     6: {1,2}      26: {1,6}        45: {2,2,3}
     7: {4}        27: {2,2,2}      46: {1,9}
     9: {2,2}      29: {10}         47: {15}
    10: {1,3}      30: {1,2,3}      49: {4,4}
    11: {5}        31: {11}         50: {1,3,3}
    13: {6}        33: {2,5}        51: {2,7}
    14: {1,4}      34: {1,7}        53: {16}
    15: {2,3}      35: {3,4}        54: {1,2,2,2}
    17: {7}        36: {1,1,2,2}    55: {3,5}
    18: {1,2,2}    37: {12}         57: {2,8}
    19: {8}        38: {1,8}        58: {1,10}
    21: {2,4}      39: {2,6}        59: {17}
    22: {1,5}      41: {13}         60: {1,1,2,3}

Crossrefs

For mean instead of median we have A057716, counted by A000065.
These partitions are counted by A238495.
The complement is A364056, counted by A027336, low version A363488.
A000975 counts subsets with integer median, A051293 for mean.
A124943 counts partitions by low median, high version A124944.
A360005 gives twice the median of prime indices, A360459 for prime factors.
A359893 and A359901 count partitions by median.
Cf. A002865, A316413, A325347, A327473, A327476, A363727.

Programs

  • Mathematica
    prifacs[n_]:=If[n==1,{},Flatten[ConstantArray@@@FactorInteger[n]]];
Select[Range[100],Median[prifacs[#]]>2&]

Formula

A360005(a(n)) > 1.
A360459(a(n)) > 2.
Previous Showing 41-44 of 44 results.

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