Ant King - cp's OEIS frontend

A228857 Odd primes p > 3 for which 14*p+1 is also prime.

5, 17, 47, 53, 59, 83, 107, 113, 149, 167, 173, 239, 269, 353, 419, 443, 449, 503, 509, 563, 587, 599, 647, 659, 677, 719, 797, 827, 929, 947, 977, 983, 1097, 1103, 1109, 1187, 1193, 1223, 1229, 1259, 1289, 1367, 1409, 1427, 1433, 1439, 1493, 1523, 1667

Offset: 1

Ant King, Sep 06 2013

nonn

In 1823, Legendre proved that the first case of Fermat’s Last Theorem is true for all exponents that are members of this sequence (see Ribenboim’s reference, p.112).

			As both 5 and 14*5 + 1 = 71 are prime, then 5 is a member of this sequence.

Paulo Ribenboim; Fermat’s Last Theorem For Amateurs, Springer-Verlag, New York, Inc., (1999).

Cf. A005384, A023212, A023228, A023237, A155943.

[p: p in PrimesInInterval(5,2000) |IsPrime(14*p+1)]; // Vincenzo Librandi, Sep 18 2016

Select[Prime[Range[3,1667]],PrimeQ[14#+1] &]

lista(nn) = forprime(p=5, nn, if(isprime(14*p+1), print1(p, ", "))); \\ Altug Alkan, Sep 18 2016

A227297 Suppose that (m, m+1) is a pair of consecutive powerful numbers as defined by A001694. This sequence gives the values of m for which neither m nor m+1 are perfect squares.

Original entry on oeis.org

12167, 5425069447, 11968683934831, 28821995554247, 48689748233307, 161461422688535037152, 3887785221910670811499

Offset: 1

Ant King, Jul 07 2013

a(1) to a(5) were found by Jaroslaw Wroblewski, who also proved that this sequence is infinite (see link to Problem 53 below). However, there are no more terms less than 500^6 = 1.5625*10^16.

A subsequence of A060355 and of A001694.

			12167 is a term because (12167, 12168) are a pair of consecutive powerful numbers, neither of which are perfect squares.
235224 is not a term because although (235224, 235225) are a pair of consecutive powerful numbers, the larger member of the pair is a square number (= 485^2).

Richard K. Guy, Unsolved Problems in Number Theory, 2nd ed., New York, Springer-Verlag, (1994), pp. 70-74. (See Powerful numbers, section B16.)

Solomon W. Golomb, Powerful numbers, Amer. Math. Monthly, Vol. 77, No. 8 (October 1970), 848-852.
Carlos Rivera, Problem 53: Powerful numbers revisited, The Prime Puzzles & Problems Connection.
David T. Walker, Consecutive integer pairs of powerful numbers and related Diophantine equations, Fibonacci Quart., Vol. 14, No. 2 (1976), pp. 111-116.
Wikipedia, Powerful number.
Index entries for sequences related to powerful numbers.

Cf. A060355, A001694.

a(6)-a(7) from the b-file at A060355 added by Amiram Eldar, Mar 22 2025

A222883 Decimal expansion of Sierpiński's third constant, K3 = lim_{n->oo} ((1/n) * (Sum_{i=1..n} (A004018(i))^2) - 4* log(n)).

Original entry on oeis.org

8, 0, 6, 6, 4, 8, 6, 1, 8, 2, 9, 3, 3, 6, 3, 2, 4, 6, 1, 0, 5, 1, 1, 8, 7, 4, 3, 8, 8, 6, 0, 4, 6, 1, 7, 0, 5, 8, 0, 0, 7, 3, 6, 7, 1, 0, 0, 9, 4, 5, 8, 9, 9, 2, 2, 4, 4, 3, 6, 7, 7, 1, 3, 3, 7, 9, 1, 2, 5, 7, 3, 6, 6, 4, 6, 4, 7, 3, 1, 1, 4, 9, 0, 2, 1, 6, 5, 4, 0, 5, 5, 9, 3, 2, 2, 4, 7, 2, 1, 6, 7, 8, 1, 5, 1

Offset: 1

Ant King, Mar 11 2013

Sierpiński introduced three constants in his 1908 doctoral thesis. The first, K, is very well known, bears his name and its decimal expansion is given in A062089. However, the second and third of these constants appear to have been largely forgotten. This sequence gives the decimal expansion of the third one, K3, and A222882 gives the decimal expansion of the second one, K2. The formula given below show that K3 is related to several other, naturally occurring constants including K and K2.

			K3 = 8.066486182933632461051187438860461705800736710094589922443677...

Steven R. Finch, Mathematical Constants, Encyclopaedia of Mathematics and its Applications, Cambridge University Press (2003), p.123. Corrigenda in the link below.

G. C. Greubel, Table of n, a(n) for n = 1..10000
Steven R. Finch, Errata and Addenda to Mathematical Constants, (June 2012), pp. 15-16.
A. Schinzel, Wacław Sierpiński’s papers on the theory of numbers, Acta Arithmetica XXI, (1972), pp. 7-13. Corrigenda in "Dzieje Matematyki Polskiej" (Wrocław 2012), p.228 (in Polish).

Cf. A001620, A004018, A014549, A062089, A062539, A074962, A222882.

Take[RealDigits[N[4/3 (24*Log[Gamma[3/4]] - 12*Log[Pi] + 72*Log[Glaisher] - 5*Log[2] + 6*EulerGamma - 3), 100]][[1]], 86]

4*log(exp(5*Euler-1)/(2^(5/3)/agm(sqrt(2),1)^4))-48/Pi^2*zeta'(2) - 4*Euler \\ Charles R Greathouse IV, Dec 12 2013

K3 = 8*K / Pi - 48 / Pi^2 * zeta'(2) + 4 * log(2) / 3 - 4, where K  is Sierpinski's first constant (A062089).
K3 = 4 / 3 * log(A^72 * e^(6 * eulergamma - 3)*( Gamma(3/4))^24 / (32 * pi^12)), where A is the Glaisher-Kinkelin constant (A074962) and eulergamma is the Euler-Mascheroni constant (A001620).
K3 = 4*log(exp(5*eulergamma - 1) / (2^(5 / 3) * G^4)) - 48 / Pi^2 * zeta'(2) - 4* eulergamma, where G is Gauss’ AGM constant (A014549).
K3 = 4*log(Pi^4 * e^(5*eulergamma - 1) / (2^(5 / 3) * L^4)) - 48 / Pi^2 * zeta'(2) - 4* eulergamma, where L is Gauss’ lemniscate constant (A062539).
K3 = 4*K / Pi + Pi * K2 - 4 * eulergamma, where K2  is Sierpiński's second constant (A222882).
1 / 4 * K3 - 1 / 4 * Pi * K2 - log(pi^2 / (2 * L^2)) = eulergamma.
1 / 4 * K3 - 1 / 4 * Pi * K2 + log(2 * G^2) = eulergamma.

More terms from Robert G. Wilson v, Oct 19 2013

A222882 Decimal expansion of Sierpiński's second constant, K2 = lim_{n->oo} ((1/n) * (Sum_{i=1..n} A004018(i^2)) - 4/Pi * log(n)).

Original entry on oeis.org

2, 2, 5, 4, 9, 2, 2, 4, 6, 2, 8, 8, 8, 2, 6, 4, 7, 6, 6, 2, 6, 8, 1, 8, 4, 7, 5, 9, 5, 2, 8, 7, 2, 3, 5, 5, 7, 8, 7, 1, 6, 6, 1, 5, 9, 8, 6, 0, 5, 3, 5, 1, 8, 8, 9, 1, 3, 8, 3, 1, 1, 6, 1, 8, 8, 5, 9, 1, 7, 2, 9, 2, 8, 9, 5, 9, 7, 1, 3, 9, 3, 4, 1, 0, 5, 8

Offset: 1

Ant King, Mar 11 2013

Sierpiński introduced three constants in his 1908 doctoral thesis. The first, K, is very well known, bears his name and its decimal expansion is given in A062089. However, the second and third of these constants appear to have been largely forgotten. This sequence gives the decimal expansion of the second one, K2, and A222883 gives the decimal expansion of the third , K3. The formula given below show that K2 is related to several other, naturally occurring constants.

			K2 = 2.25492246288826476626818475952872355787166159860535188913831...

Steven R. Finch, Mathematical Constants, Encyclopaedia of Mathematics and its Applications, Cambridge University Press (2003), p.123. Corrigenda in the link below.

G. C. Greubel, Table of n, a(n) for n = 1..10000
Steven R. Finch, Errata and Addenda to Mathematical Constants, (June 2012), pp. 15-16.
A. Schinzel, Wacław Sierpiński’s papers on the theory of numbers, Acta Arithmetica XXI, (1972), pp. 7-13. Corrigenda in "Dzieje Matematyki Polskiej" (Wrocław 2012), p.228 (in Polish).

Cf. A001620, A004018, A014549, A062089, A062539, A074962, A222883.

Take[Flatten[RealDigits[N[4(12 Log[Gamma[3/4]]-9 Log[Pi]+72 Log[Glaisher]-5 Log[2]+3 EulerGamma-3)/(3 Pi),100]]],86]

4/Pi*(log(exp(3*Euler-1)/(2^(2/3)/agm(sqrt(2),1)^2)) - 12/Pi^2*zeta'(2)) \\ Charles R Greathouse IV, Dec 12 2013

K2 = 4 / Pi * (eulergamma + K / Pi - 12 / Pi^2 * zeta'(2) + log(2) / 3 -1), where K  is Sierpiński's first constant (A062089) and eulergamma is the Euler-Mascheroni constant (A001620).
K2 = 4 * (12 * log(Gamma(3/4)) - 9*log(Pi) + 72*log(A) - 5*log(2) + 3 * eulergamma - 3) / (3 * Pi), where A is the Glaisher-Kinkelin constant (A074962).
K2 = 4 * (12 * log(Gamma(3/4)) + log(A^72 * e^(3*eulergamma - 3) / (32 * Pi^9))) / (3 * Pi).
K2 = 4 / Pi * (log(e^(3*eulergamma - 1) / (2^(2/3) * G^2)) - 12 / Pi^2 * zeta'(2)), where G is Gauss’ AGM constant (A014549).
K2 = 4 / Pi * (log(Pi^2 * e^(3*eulergamma - 1) / (2^(2/3) * L^2)) - 12 / Pi^2 * zeta'(2)), where L is Gauss’ lemniscate constant (A062539).

Minor edits by Vaclav Kotesovec, Nov 14 2014

A218331 Even, nonzero decagonal pyramidal numbers.

Original entry on oeis.org

38, 90, 476, 708, 1826, 2366, 4600, 5576, 9310, 10850, 16468, 18700, 26586, 29638, 40176, 44176, 57750, 62826, 79820, 86100, 106898, 114510, 139496, 148568, 178126, 188786, 223300, 235676, 275530, 289750, 335328, 351520, 403206, 421498, 479676, 500196

Offset: 1

Ant King, Oct 29 2012

			The sequence of nonzero decagonal pyramidal numbers begins 1, 11, 38, 90, 175, 301, 476, 708, 1005, 1375,... As the third even term is 476, then a(3) = 476.

Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1).

Cf. A007585, A218330.

LinearRecurrence[{1,3,-3,-3,3,1,-1},{38,90,476,708,1826,2366,4600},36]

a(n) = a(n-1) + 3*a(n-2) - 3*a(n-3) - 3*a(n-4) + 3*a(n-5) + a(n-6) - a(n-7).
a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6) + 512.
a(n) = (16*n-4*(-1)^n-1)*(4*n-(-1)^n+3)*(4*n-(-1)^n+1)/24.
G. f. 2*x*(19+26*x+136*x^2+38*x^3+37*x^4)/((1-x)^4*(1+x)^3).

A218330 Odd decagonal pyramidal numbers.

Original entry on oeis.org

1, 11, 175, 301, 1005, 1375, 3003, 3745, 6681, 7923, 12551, 14421, 21125, 23751, 32915, 36425, 48433, 52955, 68191, 73853, 92701, 99631, 122475, 130801, 158025, 167875, 199863, 211365, 248501, 261783, 304451, 319641, 368225, 385451, 440335, 459725, 521293

Offset: 1

Ant King, Oct 29 2012

nonn

			The sequence of decagonal pyramidal numbers A007585 begins 0, 1, 11, 38, 90, 175, 301, 476, 708, 1005, 1375,... As the third odd term is 175, then a(3) = 175.

Index entries for linear recurrences with constant coefficients, signature (1, 3, -3, -3, 3, 1, -1).

Cf. A007585, A218331.

LinearRecurrence[{1,3,-3,-3,3,1,-1}, {1,11,175,301,1005,1375,3003}, 37]

a(n) = a(n-1) + 3*a(n-2) - 3*a(n-3) - 3*a(n-4) + 3*a(n-5) + a(n-6) - a(n-7).
a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6) + 512.
a(n) = (16*n-4*(-1)^n-17)*(4*n-(-1)^n-3)*(4*n-(-1)^n-1)/24.
G. f. x*(1+10*x+161*x^2+96*x^3+215*x^4+22*x^5+7*x^6)/((1-x)^4*(1+x)^3).

A218329 Even 9-gonal (nonagonal) pyramidal numbers.

Original entry on oeis.org

10, 34, 80, 266, 420, 624, 1210, 1606, 2080, 3290, 4040, 4896, 6954, 8170, 9520, 12650, 14444, 16400, 20826, 23310, 25984, 31930, 35216, 38720, 46410, 50610, 55056, 64714, 69940, 75440, 87290, 93654, 100320, 114586, 122200, 130144, 147050, 156026, 165360

Offset: 1

Ant King, Oct 28 2012

nonn

			The sequence of 9-gonal (nonagonal) pyramidal numbers A007584 begins 1, 10, 34, 80, 155, 266, 420, 624, 885, 1210,.... As the third even term is 80, then a(3) = 80.

Index entries for linear recurrences with constant coefficients, signature (1, 0, 3, -3, 0, -3, 3, 0, 1, -1).

Cf. A007584, A218328.

LinearRecurrence[{1,0,3,-3,0,-3,3,0,1,-1},{10,34,80,266,420,624,1210,1606,2080,3290},39]

a(n) = a(n-1) + 3*a(n-3) - 3*a(n-4) - 3*a(n-6) + 3*a(n-7) + a(n-9) - a(n-10).
a(n) = 3*a(n-3) - 3*a(n-6) + a(n-9) + 448.
a(n) = phi(n)*(phi(n)+9)*(7*phi(n)-36)/4374, where phi(n) = 3 + 12*n - 3*cos(2*n*Pi/3) + sqrt(3)*sin(2*n*Pi/3).
G.f.: 2*x*(5+12*x+23*x^2+78*x^3+41*x^4+33*x^5+29*x^6+3*x^7)/((1-x)^4*(1+x+x^2)^3).

A218328 Odd 9-gonal (nonagonal) pyramidal numbers.

Original entry on oeis.org

1, 155, 885, 2639, 5865, 11011, 18525, 28855, 42449, 59755, 81221, 107295, 138425, 175059, 217645, 266631, 322465, 385595, 456469, 535535, 623241, 720035, 826365, 942679, 1069425, 1207051, 1356005, 1516735, 1689689, 1875315, 2074061, 2286375, 2512705, 2753499

Offset: 1

Ant King, Oct 28 2012

			The sequence of 9-gonal (nonagonal) pyramidal numbers A007584 begins 1, 10, 34, 80, 155, 266, 420, 624, 885, 1210, .... As the third odd term is 885, then a(3) = 885.

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A007584, A218329.

LinearRecurrence[{4,-6,4,-1},{1,155,885,2639},33]

a(n)=(2*n-1)*(4*n-3)*(28*n-25)/3 \\ Charles R Greathouse IV, Oct 18 2022

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + 448.
a(n) = (2*n-1)*(4*n-3)*(28*n-25)/3.
G.f.: x*(1+151*x+271*x^2+25*x^3)/(1-x)^4.
E.g.f.: 25 + exp(x)*(224*x^3 + 192*x^2 + 78*x - 75)/3. - Elmo R. Oliveira, Aug 24 2025

A218327 Even octagonal pyramidal numbers (A002414).

Original entry on oeis.org

30, 70, 364, 540, 1386, 1794, 3480, 4216, 7030, 8190, 12420, 14100, 20034, 22330, 30256, 33264, 43470, 47286, 60060, 64780, 80410, 86130, 104904, 111720, 133926, 141934, 167860, 177156, 207090, 217770, 252000, 264160, 302974, 316710, 360396, 375804, 424650

Offset: 1

Ant King, Oct 27 2012

nonn

			The sequence of octagonal pyramidal numbers A002414 begins 1, 9, 30, 70, 135, 231, 364, 540, 765, 1045, … As the third even term is 364, then a(3) = 364.

Index entries for linear recurrences with constant coefficients, signature (1, 3, -3, -3, 3, 1, -1).

Cf. A002414, A218326.

LinearRecurrence[{1,3,-3,-3,3,1,-1},{30,70,364,540,1386,1794,3480},37]

a(n) = a(n-1) + 3*a(n-2) - 3*a(n-3) - 3*a(n-4) + 3*a(n-5) + a(n-6) - a(n-7)
a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6) + 384
a(n) = (4*n-(-1)^n+1)*(4*n-(-1)^n+3)*(4*n-(-1)^n)/8
G. f. 2*x(15+20*x+102*x^2+28*x^3+27*x^4)/((1-x)^4*(1+x)^3)

A218326 Odd octagonal pyramidal numbers.

Original entry on oeis.org

1, 9, 135, 231, 765, 1045, 2275, 2835, 5049, 5985, 9471, 10879, 15925, 17901, 24795, 27435, 36465, 39865, 51319, 55575, 69741, 74949, 92115, 98371, 118825, 126225, 150255, 158895, 186789, 196765, 228811, 240219, 276705, 289641, 330855, 345415, 391645, 407925

Offset: 1

Ant King, Oct 27 2012

nonn

			The sequence of octagonal pyramidal numbers A002414 begins 1, 9, 30, 70, 135, 231, 364, 540, 765, 1045, … As the third odd term is 135, then a(3) = 135.

Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1).

Cf. A002414, A218327.

LinearRecurrence[{1,3,-3,-3,3,1,-1},{1,9,135,231,765,1045,2275},38]

a(n) = a(n-1) + 3*a(n-2) - 3*a(n-3) - 3*a(n-4) + 3*a(n-5) + a(n-6) - a(n-7).
a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6) + 384.
a(n) = (4*n-(-1)^n-1)*(4*n-(-1)^n-3)*(4*n-(-1)^n-4)/8.
G. f. x(1+8*x+123*x^2+72*x^3+159*x^4+16*x^5+5*x^6)/((1-x)^4*(1+x)^3).

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User: Ant King

A228857 Odd primes p > 3 for which 14*p+1 is also prime.

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A227297 Suppose that (m, m+1) is a pair of consecutive powerful numbers as defined by A001694. This sequence gives the values of m for which neither m nor m+1 are perfect squares.

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A222883 Decimal expansion of Sierpiński's third constant, K3 = lim_{n->oo} ((1/n) * (Sum_{i=1..n} (A004018(i))^2) - 4* log(n)).

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A222882 Decimal expansion of Sierpiński's second constant, K2 = lim_{n->oo} ((1/n) * (Sum_{i=1..n} A004018(i^2)) - 4/Pi * log(n)).

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A218331 Even, nonzero decagonal pyramidal numbers.

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A218330 Odd decagonal pyramidal numbers.

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A218329 Even 9-gonal (nonagonal) pyramidal numbers.

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