A220505 -id:A220505 - cp's OEIS frontend

A220485 spt(5n+4) where spt(n) = A092269(n).

10, 80, 440, 1820, 6545, 20630, 59960, 161840, 412950, 1002435, 2335760, 5246120, 11416820, 24146510, 49795175, 100348950, 198063060, 383516700, 729726660, 1366124700, 2519441030, 4581865140, 8224627160, 14584074770, 25565740130, 44334556890, 76102268520

Offset: 0

Omar E. Pol, Jan 18 2013

nonn

a(n) is divisible by 5 (see A220505).

G. E. Andrews, The number of smallest parts in the partitions of n

Cf. A092269, A019897, A213260, A220502, A220503, A220505.

a(n) = A092269(A016897(n)).

A220507 a(n) = spt(7n+5)/7 where spt(n) = A092269(n).

Original entry on oeis.org

2, 34, 260, 1498, 6956, 28024, 100953, 333680, 1026540, 2976024, 8197962, 21608760, 54788100, 134217717, 318816426, 736549424, 1659169712, 3652248590, 7870890952, 16633964444, 34522173765, 70450341042, 141526909340, 280158178412

Offset: 0

Omar E. Pol, Jan 18 2013

nonn

That spt(7n+5) == 0 (mod 7) is one of the congruences stated by George E. Andrews. See theorem 2 in the Andrews' paper. See also A220505 and A220513.

G. E. Andrews, The number of smallest parts in the partitions of n
G. E. Andrews, F. G. Garvan, and J. Liang, Combinatorial interpretation of congruences for the spt-function
K. C. Garrett, C. McEachern, T. Frederick, O. Hall-Holt, Fast computation of Andrews' smallest part statistic and conjectured congruences, Discrete Applied Mathematics, 159 (2011), 1377-1380.
F. G. Garvan, Congruences for Andrews' smallest parts partition function and new congruences for Dyson's rank
F. G. Garvan, Congruences for Andrews' spt-function modulo powers of 5, 7 and 13
F. G. Garvan, Congruences for Andrews' spt-function modulo 32760 and extension of Atkin's Hecke-type partition congruences, arXiv:1011.1957 [math.NT], 2010.
K. Ono, Congruences for the Andrews spt-function

Cf. A017041, A071746, A092269, A220502, A220505, A220513.

b[n_, i_] := b[n, i] = If[n==0 || i==1, n, {q, r} = QuotientRemainder[n, i]; If[r == 0, q, 0] + Sum[b[n - i*j, i - 1], {j, 0, n/i}]];
spt[n_] := b[n, n];
a[n_] := spt[7 n + 5]/7;
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Jan 30 2019, after Alois P. Heinz in A092269 *)

a(n) = A092269(A017041(n))/7 = A220502(n)/7.

A220513 a(n) = spt(13n+6)/13 where spt(n) = A092269(n).

Original entry on oeis.org

2, 140, 3042, 38054, 344212, 2488260, 15235620, 81926240, 396603536, 1759312286, 7246532360, 27998586490, 102294344881, 355704104008, 1183463874068, 3784162891544, 11672177600660, 34840196162760, 100912078549712, 284295561826160

Offset: 0

Omar E. Pol, Jan 18 2013

nonn

That spt(13n+6) == 0 (mod 13) is one of the congruences stated by George E. Andrews. See theorem 2 in the Andrews' paper. See also A220505 and A220507.

G. E. Andrews, The number of smallest parts in the partitions of n
G. E. Andrews, F. G. Garvan, and J. Liang, Combinatorial interpretation of congruences for the spt-function
K. C. Garrett, C. McEachern, T. Frederick, O. Hall-Holt, Fast computation of Andrews' smallest part statistic and conjectured congruences, Discrete Applied Mathematics, 159 (2011), 1377-1380.
F. G. Garvan, Congruences for Andrews' smallest parts partition function and new congruences for Dyson's rank
F. G. Garvan, Congruences for Andrews' spt-function modulo powers of 5, 7 and 13
F. G. Garvan, Congruences for Andrews' spt-function modulo 32760 and extension of Atkin's Hecke-type partition congruences, arXiv:1011.1957 [math.NT], 2010.
K. Ono, Congruences for the Andrews spt-function

Cf. A076394, A092269, A186113, A220503, A220505, A220507.

b[n_, i_] := b[n, i] = If[n==0 || i==1, n, {q, r} = QuotientRemainder[n, i]; If[r == 0, q, 0] + Sum[b[n - i*j, i - 1], {j, 0, n/i}]];
spt[n_] := b[n, n];
a[n_] := spt[13 n + 6]/13;
Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Jan 30 2019, after Alois P. Heinz in A092269 *)

a(n) = A092269(A186113(n))/13 = A220503(n)/13.

cp's OEIS Frontend

A220485 spt(5n+4) where spt(n) = A092269(n).

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A220507 a(n) = spt(7n+5)/7 where spt(n) = A092269(n).

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Formula

A220513 a(n) = spt(13n+6)/13 where spt(n) = A092269(n).

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