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 11-18 of 18 results.

A067463 Primes p such that p+sigma(p+1) is prime.

Original entry on oeis.org

5, 13, 19, 23, 29, 37, 41, 43, 53, 59, 61, 67, 83, 101, 103, 113, 131, 149, 157, 163, 167, 193, 223, 227, 229, 239, 263, 271, 281, 283, 293, 311, 313, 331, 347, 349, 373, 397, 401, 433, 461, 467, 491, 503, 523, 563, 571, 599, 601, 607, 631, 643, 653, 661, 683
Offset: 1

Views

Author

Benoit Cloitre, Feb 23 2002

Keywords

Links

Crossrefs

Cf. A008333, A067464.

Programs

  • Mathematica
    Select[Prime[Range[124]], PrimeQ[# + DivisorSigma[1, #+1]] &] (* Amiram Eldar, Apr 24 2025 *)
  • PARI
    isok(p) = isprime(p) && isprime(p+sigma(p+1)); \\ Michel Marcus, Feb 17 2021

A280555 Primes p such that sigma(sigma(p)) is a Fibonacci number.

Original entry on oeis.org

19289, 7391381, 9041581, 9124081, 9589141, 645617593711, 786881099503, 793374393583, 188950298985689, 215446003400539, 228846950929339, 257138974382029, 265666386165589, 276918720321829, 280481623844131, 323331286115017, 326905876894417
Offset: 1

Views

Author

Altug Alkan, Jan 05 2017

Keywords

Comments

Is this sequence infinite?

Examples

			Prime number 7391381 is a term because sigma(sigma(7391381)) = 14930352 is a Fibonacci number.

Crossrefs

Cf. A008333, A051027, A272412, A280545.

Programs

  • PARI
    isFibonacci(n)=my(k=n^2); issquare(k+=(k+1)<<2) || (n>0 && issquare(k-8));
is(n)=isFibonacci(sigma(n+1))&&isprime(n);

Extensions

Terms confirmed by Giovanni Resta, Jan 07 2017

A326391 Lesser of twin primes p >= 3 for which sigma(p+1)/sigma(p-1) reaches record value, where sigma(n) is the divisor sum function (A000203).

Original entry on oeis.org

3, 7559, 42839, 55439, 110879, 415799, 1713599, 1940399, 2489759, 6652799, 6846839, 15855839, 31600799, 85765679, 232792559, 845404559, 1470268799, 6299092799, 10708457759, 17459441999, 32125373279, 135019684799, 439977938399, 449755225919, 1799020903679, 2126560035599, 2835413380799, 6278415343199
Offset: 1

Views

Author

Amiram Eldar, Sep 11 2019

Keywords

Comments

Garcia et al. proved that assuming Dickson's conjecture, {sigma(p+1)/sigma(p-1) : p and p+2 are prime} is dense in [0, oo), and thus this sequence is infinite.

Examples

			The values of sigma(p+1)/sigma(p-1) for the first terms are 2.333... < 2.539... < 2.621... < 2.734... < 2.836...

Links

Crossrefs

Cf. A000203, A001359, A006512, A008332, A008333, A326356.

Programs

  • Mathematica
    s = {}; rm = 0; p = 2; Do[q = NextPrime[p]; If[q - p != 2, p = q; Continue[]]; r = DivisorSigma[1, p + 1]/DivisorSigma[1, p - 1]; If[r > rm, rm = r; AppendTo[s, p]]; p = q, {10^6}]; s

Extensions

a(22)-a(28) from Giovanni Resta, Nov 01 2019

A326392 Lesser of twin primes p for which sigma(p+1)/sigma(p) reaches record value, where sigma(n) is the divisor sum function (A000203).

Original entry on oeis.org

3, 5, 11, 29, 59, 179, 239, 419, 1319, 3119, 3359, 7559, 21839, 35279, 42839, 55439, 110879, 415799, 1713599, 1867319, 1912679, 1940399, 2489759, 3991679, 6652799, 6846839, 11531519, 28828799, 85765679, 232792559, 845404559, 1470268799, 6285399119, 6299092799
Offset: 1

Views

Author

Amiram Eldar, Sep 11 2019

Keywords

Comments

Garcia et al. proved that assuming Dickson's conjecture, {sigma(p+1)/sigma(p) : p and p+2 are prime} is dense in [2, oo), and thus this sequence is infinite.

Examples

			The values of sigma(p+1)/sigma(p) for the first terms are 1.75 < 2 < 2.333 < 2.4 < 2.8 < ...

Links

Crossrefs

Cf. A000203, A001359, A006512, A008333.

Programs

  • Mathematica
    s = {}; rm = 0; p = 2; Do[q = NextPrime[p]; If[q - p != 2, p = q; Continue[]]; r = DivisorSigma[1, p + 1]/DivisorSigma[1, p]; If[r > rm, rm = r; AppendTo[s, p]]; p = q, {10^3}]; s

A385705 Primes p such that there exists prime q < p such that sigma(p+1)=sigma(q+1).

Original entry on oeis.org

37, 61, 109, 139, 157, 181, 193, 233, 269, 283, 347, 349, 353, 367, 373, 379, 487, 521, 541, 563, 571, 593, 613, 617, 619, 641, 643, 709, 727, 739, 797, 811, 823, 829, 853, 857, 877, 907, 983, 991, 1033, 1051, 1097, 1103, 1117, 1193, 1217, 1229, 1231, 1237
Offset: 1

Views

Author

S. I. Dimitrov, Jul 07 2025

Keywords

Examples

			(41, 61) is such a pair because sigma(41+1)=sigma(61+1) = 96.

Links

Crossrefs

Cf. A000203, A000040, A008333, A385586 (a subsequence).

Programs

  • Mathematica
    s={};Do[Do[If[DivisorSigma[1,Prime[m]+1]==DivisorSigma[1,Prime[n]+1],AppendTo[s,Prime[n]];Break[]],{m,n-1}],{n,203}];s (* James C. McMahon, Jul 08 2025 *)
  • PARI
    isok(p) = my(s=sigma(p+1)); forprime(q=1, p-1, if (sigma(q+1)==s, return(q))); \\ Michel Marcus, Jul 07 2025

A385739 Primes p such that there exists a prime q < p such that sigma(q-1) = sigma(p+1) = p + q.

Original entry on oeis.org

5563, 203431, 389923, 901423, 5495263, 7418863, 28128367, 188953969, 210627577, 392753209, 402877087, 505757683, 619418689, 2549153611, 2580356851, 3953660383, 5692944349, 6806206831, 6894059071, 7082199673, 10058113363, 11307503629, 12601725943, 12615171649
Offset: 1

Views

Author

S. I. Dimitrov, Jul 08 2025

Keywords

Comments

The primes q and p form a P(-1, 1)-amicable pair. Apparently (q-1, p+1) is an amicable pair A259180.

Examples

			(5021, 5563) is such a pair because sigma(5021-1) = sigma(5563+1) = 5021 + 5563.

Links

Crossrefs

Cf. A000040, A000203, A063990, A008333, A259180, A385586, A385718.

Extensions

a(5)-a(13) from Michel Marcus, Jul 08 2025
a(14)-a(24) from Giorgos Kalogeropoulos, Jul 14 2025

A334210 a(n) = sigma(prime(n) + 1) - sigma(prime(n)).

Original entry on oeis.org

1, 3, 6, 7, 16, 10, 21, 22, 36, 42, 31, 22, 54, 40, 76, 66, 108, 34, 58, 123, 40, 106, 140, 144, 73, 114, 106, 172, 106, 126, 127, 204, 150, 196, 222, 148, 82, 130, 312, 186, 366, 154, 316, 100, 270, 265, 166, 280, 332, 202, 312, 504, 157, 476, 270, 456, 450, 286, 142, 294
Offset: 1

Views

Author

Bernard Schott, Apr 18 2020

Keywords

Comments

Lim_{n->oo} a(n) = oo because a(n) > sqrt(prime(n)) [see the reference], but this sequence is not monotone increasing.
a(n) is the sum of aliquot parts of the sum of divisors of n-th prime (see Marcus's formula). - Omar E. Pol, Apr 18 2020

Examples

			As prime(6) = 13, a(6) = sigma(14) - sigma(13) = 24 - 14 = 10.

References

  • J.-M. De Koninck & A. Mercier, 1001 Problèmes en Théorie Classique des Nombres, Problème 617 pp. 82, 280, Ellipses, Paris, 2004.

Crossrefs

Cf. A000203, A001065, A008333, A008864, A051027.

Programs

  • Maple
    G:= seq(sigma(ithprime(p)+1)-sigma(ithprime(p)), p=1..200);
  • Mathematica
    (DivisorSigma[1, # + 1] - # - 1)& @ Select[Range[300], PrimeQ] (* Amiram Eldar, Apr 18 2020 *)
  • PARI
    a(n) = my(p=prime(n)); sigma(p+1) - (p+1); \\ Michel Marcus, Apr 18 2020

Formula

a(n) = A008333(n) - A008864(n).
From Michel Marcus, Apr 18 2020: (Start)
a(n) = A001065(A008864(n)).
a(n) = A051027(prime(n)) - A000203(prime(n)). (End)

A385740 Primes p such that there exists a prime q < p such that sigma(p-1) = sigma(q-1) = p + q.

Original entry on oeis.org

1163, 7583, 17099, 48857, 65963, 172859, 5408423, 6804047, 19247087, 73162367, 77695043, 109775657, 109871933, 116464757, 160454717, 175031957, 175288493, 218543393, 268382183, 303220769, 379299989, 705800723, 823155779, 889218389, 967371143, 1100618483, 1242282407, 1701133163
Offset: 1

Views

Author

S. I. Dimitrov, Jul 08 2025

Keywords

Comments

The primes q and p form a P(-1, -1)-amicable pair. See Dimitrov link.

Examples

			(853, 1163) is such a pair because sigma(853-1) = sigma(1163-1) = 853 + 1163.

Links

Crossrefs

Cf. A000040, A000203, A063990, A008333, A259180, A385586, A385718, A385739.

Extensions

a(6)-a(25) from Michel Marcus, Jul 08 2025
a(26)-a(28) from Giorgos Kalogeropoulos, Jul 14 2025
Previous Showing 11-18 of 18 results.

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