A008864 -id:A008864 - cp's OEIS frontend

A328464 Square array A(n,k) = A276156((2^(n-1)) * (2k-1)) / A002110(n-1), read by descending antidiagonals.

1, 3, 1, 7, 4, 1, 9, 16, 6, 1, 31, 19, 36, 8, 1, 33, 106, 41, 78, 12, 1, 37, 109, 386, 85, 144, 14, 1, 39, 121, 391, 1002, 155, 222, 18, 1, 211, 124, 421, 1009, 2432, 235, 324, 20, 1, 213, 1156, 426, 1079, 2443, 4200, 341, 438, 24, 1, 217, 1159, 5006, 1086, 2575, 4213, 7430, 457, 668, 30, 1, 219, 1171, 5011, 17018, 2586, 4421, 7447, 12674, 691, 900, 32, 1

Offset: 1

Antti Karttunen, Oct 16 2019

Array is read by falling antidiagonals with n (row) and k (column) ranging as: (n,k) = (1,1), (1,2), (2,1), (1,3), (2,2), (3,1), ...

Row n contains all such sums of distinct primorials whose least significant summand is A002110(n-1), with each sum divided by that least significant primorial, which is also the largest primorial which divides that sum.

			Top left 9 X 11 corner of the array:
1: | 1,  3,   7,   9,    31,    33,    37,    39,    211,    213,    217
2: | 1,  4,  16,  19,   106,   109,   121,   124,   1156,   1159,   1171
3: | 1,  6,  36,  41,   386,   391,   421,   426,   5006,   5011,   5041
4: | 1,  8,  78,  85,  1002,  1009,  1079,  1086,  17018,  17025,  17095
5: | 1, 12, 144, 155,  2432,  2443,  2575,  2586,  46190,  46201,  46333
6: | 1, 14, 222, 235,  4200,  4213,  4421,  4434,  96578,  96591,  96799
7: | 1, 18, 324, 341,  7430,  7447,  7753,  7770, 215442, 215459, 215765
8: | 1, 20, 438, 457, 12674, 12693, 13111, 13130, 392864, 392883, 393301
9: | 1, 24, 668, 691, 20678, 20701, 21345, 21368, 765050, 765073, 765717

Cf. A328463 (transpose).
Cf. A000265, A002110, A007814, A135764, A276154, A276156,
Rows 1 - 5: A328462, A328465, A328466, A328467, A328468.
Column 2: A008864.
Column 3: A023523 (after its initial term).
Column 4: A286624.
Cf. also arrays A276945, A286625.

up_to = 105;
A002110(n) = prod(i=1,n,prime(i));
A276156(n) = { my(p=2,pr=1,s=0); while(n,if(n%2,s += pr); n >>= 1; pr *= p; p = nextprime(1+p)); (s); };
A328464sq(n,k) = (A276156((2^(n-1)) * (k+k-1)) / A002110(n-1));
A328464list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A328464sq(col,(a-(col-1))))); (v); };
v328464 = A328464list(up_to);
A328464(n) = v328464[n];

A(n,k) = A276156((2^(n-1)) * (2k-1)) / A002110(n-1).

a(n) = A328461(A135764(n)). [When all sequences are considered as one-dimensional]

A373669 Least k such that the k-th maximal run of non-prime-powers has length n. Position of first appearance of n in A110969, and the sequence ends if there is none.

Original entry on oeis.org

1, 5, 7, 12, 18, 190, 28, 109, 40, 28195574, 53

Offset: 1

Gus Wiseman, Jun 14 2024

A run of a sequence (in this case A361102) is an interval of positions at which consecutive terms differ by one.
Are there only 9 terms?
From David A. Corneth, Jun 14 2024: (Start)
No. a(10) exists.
Between the prime 144115188075855859 and 144115188075855872 = 2^57 there are 12 non-prime-powers so a(12) exists. (End)

			The maximal runs of non-prime-powers begin:
   1
   6
  10
  12
  14  15
  18
  20  21  22
  24
  26
  28
  30
  33  34  35  36
  38  39  40
  42
  44  45  46
  48
  50  51  52
  54  55  56  57  58
  60

Gus Wiseman, Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers.

For composite runs we have A073051, sorted A373400, firsts of A176246.
For squarefree runs we have firsts of A120992.
For prime-powers runs we have firsts of A174965.
For prime runs we have firsts of A251092 or A175632.
For squarefree antiruns we have A373128, firsts of A373127.
For nonsquarefree runs we have A373199, firsts of A053797.
The sorted version is A373670.
For antiruns we have firsts of A373672.
For runs of non-prime-powers:
- length A110969
- min A373676
- max A373677
- sum A373678
A000961 lists the powers of primes (including 1).
A005117 lists the squarefree numbers, first differences A076259.
A013929 lists the nonsquarefree numbers, first differences A078147.
A057820 gives first differences of consecutive prime-powers, gaps A093555.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A361102 lists the non-prime-powers, without 1 A024619.
Cf. A007053, A008864, A014963, A027833, A038664, A054265, A067774, A356068, A373401, A373403, A373671.

q=Length/@Split[Select[Range[10000],!PrimePowerQ[#]&],#1+1==#2&]//Most;
spna[y_]:=Max@@Select[Range[Length[y]],SubsetQ[y,Range[#1]]&];
Table[Position[q,k][[1,1]],{k,spna[q]}]

A006022 Sprague-Grundy (or Nim) values for the game of Maundy cake on an n X 1 sheet.

Original entry on oeis.org

0, 1, 1, 3, 1, 4, 1, 7, 4, 6, 1, 10, 1, 8, 6, 15, 1, 13, 1, 16, 8, 12, 1, 22, 6, 14, 13, 22, 1, 21, 1, 31, 12, 18, 8, 31, 1, 20, 14, 36, 1, 29, 1, 34, 21, 24, 1, 46, 8, 31, 18, 40, 1, 40, 12, 50, 20, 30, 1, 51, 1, 32, 29, 63, 14, 45, 1, 52, 24, 43, 1, 67, 1, 38, 31, 58, 12, 53, 1

Offset: 1

N. J. A. Sloane

nonn

There are three equivalent formulas for a(n). Suppose n >= 2, and let p1 <= p2 <= ... <= pk be the prime factors of n, with repetition.
Theorem 1: a(1) = 0. For n >= 2, a(n) = n*s(n), where
s(n) = 1/p1 + 1/(p1*p2) + 1/(p1*p2*p3) + ... + 1/(p1*p2*...*pk).
This is implicit in Berlekamp, Conway and Guy, Winning Ways, 2 vols., 1982, pp. 28, 53.
Note that s(n) = A322034(n) / A322035(n).
David James Sycamore observed on Nov 24 2018 that Theorem 1 implies a(n) < n for all n (see comments in A322034), and also leads to a simple recurrence for a(n):
Theorem 2: a(1) = 0. For n >= 2, a(n) = p*a(n/p) + 1, where p is the largest prime factor of n.
Proof. (Th. 1 implies Th. 2) If n is a prime, Theorem 1 gives a(n) = 1 = n*a(1)+1. For a nonprime n, let n = m*p where p is the largest prime factor of n and m >= 2. From Theorem 1, a(m) = m*s(m), a(n) = q*m*(s(m) + 1/n) = q*a(m) + 1.
(Th. 2 implies Th. 1) The reverse implication is equally easy.
Theorem 2 is equivalent to the following more complicated recurrence:
Theorem 3: a(1) = 0. For n >= 2, a(n) = max_{p|n, p prime} (p*a(n/p)+1).

			For n=24, s(24) = 1/2 + 1/4 + 1/8 + 1/24 = 11/12, so a(24) = 24*11/12 = 22.

E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982, see p. 28, 53.
E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Second Edition, Vol. 1, A K Peters, 2001, pp. 27, 51.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Jonathan Blanchette and Robert Laganière, A Curious Link Between Prime Numbers, the Maundy Cake Problem and Parallel Sorting, arXiv:1910.11749 [cs.DS], 2019.

Cf. A020639, A032742, A001348, A002182, A008864.
Cf. also A027746, A306264, A306363, A322034, A322035, A322036, A322382, A328898, A333783, A332993, A333791.
Row sums of A328969.

a006022 1 = 0
a006022 n = (+ 1) $ sum $ takeWhile (> 1) $
          iterate (\x -> x `div` a020639 x) (a032742 n)
-- Reinhard Zumkeller, Jun 03 2012

P:=proc(n) local FM: FM:=ifactors(n)[2]: seq(seq(FM[j][1], k=1..FM[j][2]), j=1..nops(FM)) end: # A027746
s:=proc(n) local i,t,b; global P;t:=0; b:=1; for i in [P(n)] do b:=b*i; t:=t+1/b; od; t; end; # A322034/A322035
A006022 := n -> if n = 1 then 0 else n*s(n); fi;
# N. J. A. Sloane, Nov 28 2018

Nest[Function[{a, n}, Append[a, Max@ Map[# a[[n/#]] + 1 &, Rest@ Divisors@ n]]] @@ {#, Length@ # + 1} &, {0, 1}, 77] (* Michael De Vlieger, Nov 23 2018 *)

lista(nn) = {my(v = vector(nn)); for (n=1, nn, if (n>1, my(m = 0); fordiv (n, d, if (d>1, m = max(m, d*v[n/d]+1))); v[n] = m;); print1(v[n], ", "););} \\ Michel Marcus, Nov 25 2018

a(n) = n * Sum_{k=1..N} (1/(p1^m1*p2^m2*...*pk^mk)) * (pk^mk-1)/(pk-1) for n>=2, where pk is the k-th distinct prime factor of n, N is the number of distinct prime factors of n, and mk is the multiplicity of pk occurring in n. To prove this, expand the factors in Theorem 1 and use the geometrical series identity. - Jonathan Blanchette, Nov 01 2019
From Antti Karttunen, Apr 12 2020: (Start)
a(n) = A322382(n) + A333791(n).
a(n) = A332993(n) - n = A001065(n) - A333783(n). (End)
a(n) = Sum_{k=1..bigomega(n)} F^k(n), where F^k(n) is the k-th iterate of F(n) = A032742(n). - Ridouane Oudra, Jan 26 2024

Edited and extended by Christian G. Bower, Oct 18 2002

Entry revised by N. J. A. Sloane, Nov 28 2018

A175222 a(n) = prime(n) + 5.

Original entry on oeis.org

7, 8, 10, 12, 16, 18, 22, 24, 28, 34, 36, 42, 46, 48, 52, 58, 64, 66, 72, 76, 78, 84, 88, 94, 102, 106, 108, 112, 114, 118, 132, 136, 142, 144, 154, 156, 162, 168, 172, 178, 184, 186, 196, 198, 202, 204, 216, 228, 232, 234, 238, 244, 246, 256, 262, 268, 274, 276

Offset: 1

Jaroslav Krizek, Mar 06 2010

a(n) = A000040(n) + 5 = A008864(n) + 4 = A052147(n) + 3 = A113395(n) + 2 = A175221 (n) + 1 = A139049(n) - 1 = A175223(n) - 2 = A175224(n) - 3 = A140353(n) - 4 = A175225(n) - 5.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000040, A008864, A052147, A086801, A113395, A113935, A139049, A140353, A175221, A175223, A175224, A175225.

Filtered([1..300], k-> IsPrime(k) ) + 5; # G. C. Greubel, May 20 2019

[p+5: p in PrimesUpTo(500)]; // Vincenzo Librandi, Dec 04 2010

Prime[Range[120]]+5 (* Vladimir Joseph Stephan Orlovsky, Nov 21 2010 *)

{a(n) = prime(n) + 5}; \\ G. C. Greubel, May 20 2019

[nth_prime(n) + 5 for n in (1..100)] # G. C. Greubel, May 20 2019

More terms from Vincenzo Librandi, Mar 14 2010

A373670 Numbers k such that the k-th run-length A110969(k) of the sequence of non-prime-powers (A024619) is different from all prior run-lengths.

Original entry on oeis.org

1, 5, 7, 12, 18, 28, 40, 53, 71, 109, 170, 190, 198, 207, 236, 303, 394, 457, 606, 774, 1069, 1100, 1225, 1881, 1930, 1952, 2247, 2281, 3140, 3368, 3451, 3493, 3713, 3862, 4595, 4685, 6625, 8063, 8121, 8783, 12359, 12650, 14471, 14979, 15901, 17129, 19155

Offset: 1

Gus Wiseman, Jun 15 2024

nonn

The unsorted version is A373669.

			The maximal runs of non-prime-powers begin:
   1
   6
  10
  12
  14  15
  18
  20  21  22
  24
  26
  28
  30
  33  34  35  36
  38  39  40
  42
  44  45  46
  48
  50  51  52
  54  55  56  57  58
  60
So the a(n)-th runs begin:
   1
  14  15
  20  21  22
  33  34  35  36
  54  55  56  57  58

For nonsquarefree runs we have A373199 (if increasing), firsts of A053797.
For squarefree antiruns see A373200, unsorted A373128, firsts of A373127.
For composite runs we have A373400, unsorted A073051, firsts of A176246.
For prime antiruns we have A373402.
For runs of non-prime-powers:
- length A110969, firsts A373669, sorted A373670 (this sequence):
- min A373676
- max A373677
- sum A373678
For runs of prime-powers:
- length A174965
- min A373673
- max A373674
- sum A373675
A000961 lists the powers of primes (including 1).
A057820 gives first differences of consecutive prime-powers, gaps A093555.
A361102 lists the non-prime-powers, without 1 A024619.
Cf. A005381, A008864, A014963, A027833, A038664, A356068, A373401, A373574.

t=Length/@Split[Select[Range[10000],!PrimePowerQ[#]&],#1+1==#2&];
Select[Range[Length[t]],FreeQ[Take[t,#-1],t[[#]]]&]

A023512 Exponent of 2 in prime factorization of prime(n) + 1.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling

2^a(n) is the largest power of 2 dividing (prime(n)+1).

By Dirichlet's theorem on arithmetic progressions, the asymptotic density of primes p such that p == 2^k-1 (mod 2^k) within all the primes is 1/2^(k-1), for k >= 1. This is also the asymptotic density of terms in this sequence that are >= k. Therefore, the asymptotic density of the occurrences of k in this sequence is d(k) = 1/2^(k-1) - 1/2^k = 1/2^k, and the asymptotic mean of this sequence is Sum_{k>=1} k*d(k) = 2. - Amiram Eldar, Mar 14 2025

			a(9) = 3 because the 9th prime is 23 and the largest power of 2 dividing 24 is 3.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..2261 from K. G. Stier)

Cf. A007814, A008864, A023506, A239114.

[Valuation(NthPrime(n)+1, 2): n in [1..110]]; // Bruno Berselli, Aug 05 2013

with(numtheory): a:=proc(n) local div,s,j,c: div:=divisors(1+ithprime(n)): s:=nops(div): for j from 1 to s do if type(simplify(log[2](div[j])), integer)=true then c[j]:=simplify(log[2](div[j])) else c[j]:=0 fi od: max(seq(c[j],j=1..s)) end: seq(a(n),n=1..120); # most probably not the simplest Maple program - Emeric Deutsch, Jul 20 2005

Join[{0}, Table[FactorInteger[Prime[n] + 1][[1]][[2]], {n, 2, 100}]] (* Clark Kimberling, Oct 01 2013 *)
IntegerExponent[Prime[Range[100]] + 1, 2] (* Zak Seidov, Apr 25 2014 *)

a(n)=valuation(prime(n)+1,2);
vector(100,n,a(n)) \\ Joerg Arndt, Mar 11 2014

a(n) = A007814(A008864(n)). - Amiram Eldar, Mar 14 2025

Corrected by Yasutoshi Kohmoto, Feb 25 2005

Edited by N. J. A. Sloane, Dec 23 2006

A131992 a(n) = 1 + prime(n) + prime(n)^2 + prime(n)^3 + prime(n)^4.

Original entry on oeis.org

31, 121, 781, 2801, 16105, 30941, 88741, 137561, 292561, 732541, 954305, 1926221, 2896405, 3500201, 4985761, 8042221, 12326281, 14076605, 20456441, 25774705, 28792661, 39449441, 48037081, 63455221, 89451461, 105101005, 113654321

Offset: 1

Reinhard Zumkeller, Aug 06 2007

Thébault shows that a(2) = 121 is the only square in this sequence. - Charles R Greathouse IV, Jul 23 2013

Giovanni Resta has found that 28792661 is the first Sophie Germain prime of this form (and actually of the form p = (n^m-1)/(n-1) for any p-1 > n, m > 1). - M. F. Hasler, Mar 03 2020

			a(1) = 31 because prime(1) = 2 and 1 + 2 + 2^2 + 2^3 + 2^4 = 1 + 2 + 4 + 8 + 16 = 31.

Victor Thébault, Curiosités arithmétiques, Mathesis 62 (1953), pp. 120-129.

Ivan Panchenko, Table of n, a(n) for n = 1..1000

Cf. A030514, A008864, A060800, A131993.

Equals A053699 restricted to prime indices. Subsequence of primes is A190527.

Table[Sum[Prime[n]^k, {k, 0, 4}], {n, 30}] (* Alonso del Arte, May 24 2015 *)

a(n)=sigma(prime(n)^4)  \\ Charles R Greathouse IV, Jul 23 2013

a(n) = 1 + A131991(n)*A000040(n).
a(n) = (A050997(n) - 1)/A006093(n).
a(n) = A000203(prime(n)^4). - R. J. Mathar, Mar 15 2018
a(n) = (prime(n)^5 - 1)/(prime(n) - 1) = A053699(prime(n)). (This is also meant by the 2nd formula.) - M. F. Hasler, Mar 03 2020

A140353 a(n) = prime(n) + 9.

Original entry on oeis.org

11, 12, 14, 16, 20, 22, 26, 28, 32, 38, 40, 46, 50, 52, 56, 62, 68, 70, 76, 80, 82, 88, 92, 98, 106, 110, 112, 116, 118, 122, 136, 140, 146, 148, 158, 160, 166, 172, 176, 182, 188, 190, 200, 202, 206, 208, 220, 232, 236, 238, 242, 248, 250, 260, 266, 272, 278, 280, 286

Offset: 1

Odimar Fabeny, May 30 2008

a(n) = A000040(n) + 9 = A008864(n) + 8 = A052147(n) + 7 = A113395(n) + 6 = A175221(n) + 5 = A175222(n) + 4 = A139049(n) + 3 = A175223(n) + 2 = A175224(n) + 1 = A175225(n) - 1. - Jaroslav Krizek, Mar 06 2010

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A139049

Filtered([1..300], k-> IsPrime(k) ) +9 # G. C. Greubel, May 20 2019

[NthPrime(n)+9: n in [1..70]]; // G. C. Greubel, May 20 2019

9 + Prime[Range[70]] (* G. C. Greubel, May 20 2019 *)

PARI
```
A140353(n) = prime(n)+9
```

[nth_prime(n) +9 for n in (1..70)] # G. C. Greubel, May 20 2019

Edited by Michael B. Porter, Jan 28 2010

A167053 a(1)=3; for n > 1, a(n) = 1 + a(n-1) + gcd( a(n-1)*(a(n-1)+2), A073829(a(n-1)) ).

Original entry on oeis.org

3, 19, 39, 81, 165, 333, 335, 673, 1347, 1349, 1351, 1353, 1355, 1357, 1359, 2721, 2723, 2725, 2727, 5457, 5459, 5461, 5463, 5465, 5467, 5469, 10941, 10943, 10945, 10947, 21897, 21899, 21901, 21903, 21905, 21907, 21909, 43821, 43823, 43825, 43827, 43829, 43831

Offset: 1

Vladimir Shevelev, Oct 27 2009

nonn

The first differences are 16, 20, 42, etc. They are either 2 or in A075369 or in A008864, see A167054.

A proof follows from Clement's criterion of twin primes.

			a(2) = 1 + 3 + gcd(3*5, 4*(2! + 1) + 3) = 19.

E. Trost, Primzahlen, Birkhäuser-Verlag, 1953, pages 30-31.

Amiram Eldar, Table of n, a(n) for n = 1..206
P. A. Clement, Congruences for sets of primes, Amer. Math. Monthly, 56 (1949), 23-25.

Cf. A073829, A008864, A167054.

Cf. A166944, A166945, A116533, A163961, A163963, A084662, A084663, A106108, A132199, A134162, A135506, A135508, A118679, A120293.

A073829 := proc(n) n+4*((n-1)!+1) ; end proc:
A167053 := proc(n) option remember ; local aprev; if n = 1 then 3; else aprev := procname(n-1) ; 1+aprev+gcd(aprev*(aprev+2),A073829(aprev)) ; end if; end proc:
seq(A167053(n),n=1..60) ; # R. J. Mathar, Dec 17 2009

A073829[n_] := 4((n-1)! + 1) + n;
a[1] = 3;
a[n_] := a[n] = 1 + a[n-1] + GCD[a[n-1] (a[n-1] + 2), A073829[a[n-1]]];
Array[a, 60] (* Jean-François Alcover, Mar 25 2020 *)

Definition shortened and values from a(4) on replaced by R. J. Mathar, Dec 17 2009

A355927 Square array A(n, k) = sigma(A246278(n, k)), read by falling antidiagonals.

Original entry on oeis.org

3, 7, 4, 12, 13, 6, 15, 24, 31, 8, 18, 40, 48, 57, 12, 28, 32, 156, 96, 133, 14, 24, 78, 72, 400, 168, 183, 18, 31, 48, 248, 112, 1464, 252, 307, 20, 39, 121, 84, 684, 216, 2380, 360, 381, 24, 42, 124, 781, 144, 1862, 280, 5220, 480, 553, 30, 36, 104, 342, 2801, 240, 3294, 432, 7240, 720, 871, 32, 60, 56, 372, 1064, 16105, 336, 6140, 600, 12720, 960, 993, 38

Offset: 1

Antti Karttunen, Jul 22 2022

Each column is strictly monotonic.

			The top left corner of the array:
   k=  1    2    3      4    5      6    7       8      9     10    11      12
  2k=  2    4    6      8   10     12   14      16     18     20    22      24
----+--------------------------------------------------------------------------
  1 |  3,   7,  12,    15,  18,    28,  24,     31,    39,    42,   36,     60,
  2 |  4,  13,  24,    40,  32,    78,  48,    121,   124,   104,   56,    240,
  3 |  6,  31,  48,   156,  72,   248,  84,    781,   342,   372,  108,   1248,
  4 |  8,  57,  96,   400, 112,   684, 144,   2801,  1064,   798,  160,   4800,
  5 | 12, 133, 168,  1464, 216,  1862, 240,  16105,  2196,  2394,  288,  20496,
  6 | 14, 183, 252,  2380, 280,  3294, 336,  30941,  4298,  3660,  420,  42840,
  7 | 18, 307, 360,  5220, 432,  6140, 540,  88741,  6858,  7368,  576, 104400,
  8 | 20, 381, 480,  7240, 600,  9144, 640, 137561, 11060, 11430,  760, 173760,
  9 | 24, 553, 720, 12720, 768, 16590, 912, 292561, 20904, 17696, 1008, 381600,
Note: See A355941 for the corresponding numbers in A246278 at which points the value in this array divides the term immediately below.

Cf. A000203, A246278.
Cf. A008864 (column 1), A062731 (row 1).
Cf. also A341605, A355925, A355941.

up_to = 105;
A246278sq(row,col) = if(1==row,2*col, my(f = factor(2*col)); for(i=1, #f~, f[i,1] = prime(primepi(f[i,1])+(row-1))); factorback(f));
A355927sq(row,col) = sigma(A246278sq(row,col));
A355927list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A355927sq(col,(a-(col-1))))); (v); };
v355927 = A355927list(up_to);
A355927(n) = v355927[n];

A(n, k) = A000203(A246278(n, k)).

A(n, k) = A341605(n, k) * A355925(n, k).

cp's OEIS Frontend

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