A115709 -id:A115709 - cp's OEIS frontend

A114439 Indices of semiprime pentagonal numbers.

4, 5, 6, 10, 13, 14, 29, 34, 38, 41, 46, 53, 58, 73, 86, 94, 101, 106, 109, 118, 134, 149, 181, 206, 214, 218, 226, 233, 254, 274, 281, 293, 314, 326, 349, 394, 398, 401, 409, 421, 449, 454, 458, 461, 478, 538, 541, 566, 569, 613, 626, 634, 661, 673, 694

Offset: 1

Jonathan Vos Post, Feb 14 2006

P(2) = 5 is the only prime pentagonal number, all other factor as P(k) = (k/2)*(3*k-1) or k*((3*k-1)/2) and thus have at least 2 prime factors. P(k) is semiprime iff [k prime and (3*k-1)/2 prime] or [k/2 prime and 3*k-1 prime]. A115709 is pentagonal numbers (A000326) whose digit reversal is a semiprime (A001358).

			a(1) = 4 because P(4) = PentagonalNumber(4) = 4*(3*4 -1)/2 = 22 = 2 * 11 is semiprime.
a(2) = 5 because P(5) = 5*(3*5 -1)/2 = 35 = 5 * 7 is semiprime.
a(7) = 29 because P(29) = 29*(3*29 -1)/2 = 1247 = 29 * 43 is semiprime.
a(8) = 34 because P(34) = 34*(3*34 -1)/2 = 1717 = 17 * 101 is semiprime.
a(17) = 101 because P(101) = 101*(3*101 -1)/2 = 15251 = 101 * 151 is semiprime.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)
Eric Weisstein's World of Mathematics, Pentagonal Number.

Cf. A000326, A001222, A001358, A115709.

Position[PolygonalNumber[5,Range[700]],?(PrimeOmega[#]==2&)]//Flatten (* _Harvey P. Dale, Oct 02 2021 *)

{a(n)} = {k such that A001222(A000326(k)) = 2}.
{a(n)} = {k such that k*(3*k-1)/2 has exactly 2 prime factors}.
{a(n)} = {k such that A000326(k) is an element of A001358}.

More terms from Giovanni Resta, Jun 14 2016

A115708 Semiprimes (A001358) whose digit reversal is a pentagonal number (A000326).

Original entry on oeis.org

10, 15, 21, 22, 235, 287, 517, 529, 671, 1513, 1555, 2611, 5137, 5331, 6241, 7017, 7171, 7421, 7597, 7814, 10078, 10913, 15011, 15094, 15214, 15251, 20395, 20603, 21542, 23129, 24501, 24873, 25157, 26851, 27161, 51998, 53297, 54989, 55551

Offset: 1

Giovanni Resta, Jan 31 2006

			235=5*47 is semiprime and 532 is the 19th pentagonal number.

Harvey P. Dale, Table of n, a(n) for n = 1..100

Cf. A000326, A001358, A115709.

Select[Range[60000],PrimeOmega[#]==2&&IntegerQ[(1+Sqrt[1+24* IntegerReverse[#]])/ 6]&] (* Harvey P. Dale, Apr 27 2022 *)

A114441 Indices of 3-almost prime pentagonal numbers.

Original entry on oeis.org

3, 7, 8, 9, 17, 18, 20, 21, 22, 23, 25, 26, 28, 30, 31, 37, 44, 49, 50, 61, 62, 65, 66, 69, 71, 74, 76, 78, 79, 85, 89, 93, 97, 98, 113, 116, 121, 122, 129, 130, 133, 137, 141, 146, 148, 151, 154, 157, 158, 161, 164, 166, 170, 173, 174, 178, 185, 186, 188, 190, 193, 194

Offset: 1

Jonathan Vos Post, Feb 14 2006

P(2) = 5 is the only prime pentagonal number, all other factor as P(k) = (k/2)*(3*k-1) or k*((3*k-1)/2) and thus have at least 2 prime factors. P(k) is semiprime iff [k prime and (3*k-1)/2 prime] or [k/2 prime and 3*k-1 prime].

			a(1) = 3 because P(3) = PentagonalNumber(3) = 3*(3*3 -1)/2 = 12 = 2^2 * 3 is a 3-almost prime.
a(2) = 7 because P(7) = 7*(3*7 -1)/2 = 70 = 2 * 5 * 7 is a 3-almost prime.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Almost Prime.
Eric Weisstein's World of Mathematics, Pentagonal Number.

Cf. A000326, A001222, A014612, A115709.

A000326 := proc(n) n*(3*n-1)/2 ; end: isA014612 := proc(n) option remember ; RETURN( numtheory[bigomega](n) = 3) ; end: for n from 1 to 400 do if isA014612(A000326(n)) then printf("%d,",n) ; fi; od: # R. J. Mathar, Jan 27 2009

Select[Range[200], PrimeOmega[PolygonalNumber[5, #]] == 3 &] (* Amiram Eldar, Oct 06 2024 *)

{a(n)} = {k such that A001222(A000326(k)) = 3}.
{a(n)} = {k such that k*(3*k-1)/2 has exactly 3 prime factors}.
{a(n)} = {k such that A000326(k) is an element of A014612}.

125 removed, 145 replaced with 146 by R. J. Mathar, Jan 27 2009

A114443 Indices of 4-almost prime pentagonal numbers.

Original entry on oeis.org

12, 15, 16, 19, 24, 33, 36, 39, 45, 47, 52, 55, 56, 57, 60, 68, 70, 77, 82, 83, 84, 88, 90, 95, 102, 103, 104, 105, 110, 111, 114, 119, 124, 127, 138, 140, 142, 143, 145, 150, 153, 156, 163, 169, 172, 177, 179, 182, 183, 191, 196, 198

Offset: 1

Jonathan Vos Post, Feb 14 2006

P(2) = 5 is the only prime pentagonal number, all other factor as P(k) = (k/2)*(3*k-1) or k*((3*k-1)/2) and thus have at least 2 prime factors. P(k) is semiprime iff [k prime and (3*k-1)/2 prime] or [k/2 prime and 3*k-1 prime].

			a(1) = 12 because P(12) = A000326(12) = 12*(3*12-1)/2 = 210 = 2 * 3 * 5 * 7 is a 4-almost prime (in fact the primorial prime(4)#).
a(3) = 16 because P(16) = 16*(3*16-1)/2 = 376 = 2^3 * 47 is a 4-almost prime (the prime factors need not be distinct).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Almost Prime.
Eric Weisstein's World of Mathematics, Pentagonal Number.

Cf. A000326, A001222, A014613, A115709.

Select[Range[400], PrimeOmega[PolygonalNumber[5, #]] == 4 &] (* Amiram Eldar, Oct 06 2024 *)

{a(n)} = {k such that A001222(A000326(k)) = 4}.
{a(n)} = {k such that k*(3*k-1)/2 has exactly 4 prime factors}.
{a(n)} = {k such that A000326(k) is an element of A014613}.

82 inserted by R. J. Mathar, Dec 22 2010

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A114439 Indices of semiprime pentagonal numbers.

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A115708 Semiprimes (A001358) whose digit reversal is a pentagonal number (A000326).

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A114441 Indices of 3-almost prime pentagonal numbers.

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A114443 Indices of 4-almost prime pentagonal numbers.

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